First Edition, 2007
ISBN 978 81 89940 74 4
© All rights reserved.
Published by: Global Media 1819, Bhagirath Palace,...

Author:
Sarah Reed

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First Edition, 2007

ISBN 978 81 89940 74 4

© All rights reserved.

Published by: Global Media 1819, Bhagirath Palace, Chandni Chowk, Delhi-110 006 Email: [email protected]

Table of Contents 1. How to Read Expression Evaluation 2. Introduction to Expressions 3. Introduction to Operators and Operands 4. Unary Operators 5. Raise to a Power Operations 6. Parentheses 7. Variables and Functions 8. A Few Notes About Multiplication 9. Terms and Factors 10. Linear Functions 11. Polynomial Functions 12. Rational Functions 13. Exponential Functions 14. Trigonometry Realms 15. Intersections 16. The Geometry Section 17. Simple Data Grapher 18. Interval Notation 19. What is a Perfect Square 20. Infinity and Its Symbol

Expression Evaluation

Understanding mathematics notation is, well, essential to understanding mathematics. The material here discusses what we mean by the concept of an expression, and how one reads an expression in order to evaluate it. We will center on the concepts of operators and operands and look at expressions as if they were animated in our imagination. A few Flash animations are presented in an effort to get these ideas across. Precedence is covered here, and usually students learn that quite early in their study of mathematics. Often, though, a person is not exposed to the concept of interpreting a math expression in terms of its operators and operands. At any rate, it is suggested that you start with the first link and work your way down through the list of others. Introduction to Expressions Other Representations Besides Numbers for Values Introduction to Operators and Operands More about Operators and Operands Unary Operations Raise to a Power Operation Parentheses And Now Including Variables and Functions Just a Few Notes about Multiplication

Terms and Factors

Introduction to Expressions In a very simple manner, this certainly seems true:

As many of these circles

equals

as many of these rectangles.

So, how do we talk about and write about ‘as many of’ or ‘how many’?

One way to do it is with words. Using words, here’s how many circles there are above:

Three Mathematicians usually do not use words for these types of situations, however. They prefer other ways. Another way to describe how many circles we have here is to use digits. Here’s how many circles using digits:

3

When digits are used to describe ‘how many’ we often say we are dealing with numbers. Of course, numbers can have more than one digit, and they may include a decimal point. They can also include a positive or a negative sign, but in some discussions those are not considered part of the number, but are considered an operator on the number. Now, ‘three’ or ‘3’ is said to be the value a mathematical expression that represents ‘how many’ of the circles, (or rectangles, for that matter), that we have above. That mathematical expression, again, looks like this:

3 This mathematical expression is made up of one mathematical element, the number 3. This mathematical expression evaluates to a value of ‘three’, (if a word is used), or ‘3’, (if a number is used). Mathematical expressions can have more than one number in them. They can also contain elements that are not numbers. For example, here is another mathematical expression:

3+2 This mathematical expression evaluates to a value of ‘five’ or ‘5’. Here is another mathematical expression that evaluates to a value of ‘five’ or ‘5’:

5 Since both of these two newest expressions have the same value, they are equal expressions, and we can write:

3+2=5 Or:

5=3+2 One good way to think about a value is that it is a location on a number line. A number line is a line, drawn or imagined, with values strung along it. The value at any point can be indicated by a number called the coordinate of that point. As you move to the right along the number line the values get greater. They get less and less as you move to the left. A number line has a place called the origin; this is for the value of zero. Numbers to the right of zero are positive, numbers to the left of it negative. Here’s a picture of a number line:

So, when two expressions are equal, then each of their values is located at the same spot on the number line. Now, two expressions do not have to be equal. They could be unequal. In such a case one would find that after evaluating each expression separately to its corresponding value, then those two values would be located at different spots on the number line. So, one expression can be greater in value or lesser in value than another, but for now, let’s just deal with two expressions that are equal to each other. Other Representations Besides Numbers for Values

Now, let’s just say that there are situations where one knows that a value exists, but one does not know exactly what that value is. For example, suppose you know that there are some salt shakers in this box:

How many salt shakers that are in that box is a value. (By the way, there may be zero salt shakers in that box.) That is, somewhere along the number line there is a spot that indicates the value for the number of salt shakers. But we don’t know where that spot is. However, none of this stops us from doing mathematics with the value of the number of salt shakers in this box. We just need to understand how to represent that value. Symbols called variables are used to represent the concept of ‘how many’ when that concept is considered to exist, although the exact value of that concept may be unknown at the present, (and may continue to be unknown). Well, let’s just say that variables act like numbers, but you may not know which number a variable is acting like. Variables usually look like letters of the alphabet. So, if we want to do mathematics with the number of salt shakers in that box above, we need to decide upon a letter of the alphabet to represent the ‘how many’ for that value. Let’s let the letter ‘n’ be that letter. Then we would say: Let n be the number of salt shakers in the box. If we wish to consider that there are three salt shakers in the box, then we would make statements like:

Let n equal 3. Or: Let n = 3. Or simply: n=3 It is important to know, though, that we can do mathematics with the variable n without ever knowing how many salt shakers are in the box. For example, if the box doesn’t weigh too much, it’s possible to show that the total weight of it and its contents about

doubles if the number of salt shakers inside of it doubles. One could find out that fact mathematically without ever knowing how many salt shakers were actually in the box. So, values can be represented by numbers and variables. Numbers are made up of digits and maybe a decimal point. Sometimes one considers a positive or negative sign part of a number. The value of a number is clearly represented by itself. Its location on the number line is not up to question. Variables are symbols which are usually single letters of the alphabet. Variables are often called ‘unknowns’ because the exact value for a variable may be unknown in the problem you are facing. In fact, knowing the exact value of a variable may not actually enter into the work you are doing involving that variable. For example, this is true:

5=3+2 And this is true, too:

5+n=3+2+n The above statement is true no matter what the value for n may actually be. Here we have done some algebra using a symbol, n, for a value that is unknown. So, we now have two methods for showing a value in a mathematics expression: • Use a number, say, like 3. • Use a variable, for example, n. There is another manner in which a value can be introduced into a mathematics expression: the concept of a function. Currently, we will not go into depth regarding the definition of a function. As far as notation and algebra go, however, let’s say that functions work in mathematics expressions much like a number or a variable. The symbol for a function can have slightly different meanings in different contexts. Here, let’s say that it is a symbol which represents a value. A symbol for a function often looks like this: f(x) The above symbol is pronounced like this: ‘f of x’, (‘ef of ex’) This entire symbol, f(x), acts like a value. That is, since this is true:

5=3+2 Then so is this true: 5 + f(x) = 3 + 2 + f(x) What value does this symbol, f(x), represent? That is a good question! It’s value can change, so it acts a bit like a variable. Actually, though, its value, called its output, is

determined by what is known as its input. And the rule that ties the value of its output to the value of its input is called the definition of the function. In the symbol f(x) the x stands for the input value. The name of the function is f. And the entire symbol, f(x), stands for the output value, that is, the value that enters the mathematics expression. If we know the function definition, then we can determine the output value once we know the input value. For example, suppose the function is defined as: f(x) = 5x This states that the function will have a value, or output, equal to five times the input value. For example if 2 is input to this function, it will output 10. Starting again with the definition and placing a 2 in for the x, the math looks like this: f(x) = 5x f(2) = 5(2) f(2) = 10 For the above we would say:

‘f of two equals ten’

Of course, a different input would yield a different output. For example: f(6) = 30 For the above we would say:

‘f of six equals thirty’ Now, there is a lot more to understanding functions than this. They are mentioned here because it is important to understand the common ways in which values enter mathematics expressions right from the start when working with expression evaluation. So, now we know that there are three common ways that values are represented in mathematics expressions: •

Numbers

• •

Variables Functions

Introduction to Operators and Operands As we have seen, a mathematics expression can be made up of only one element, for example, a number, as in:

3 The above expression evaluates to three. This value of three can be represented by the number 3.

Usually, though, expressions are much more involved than the simple one above. Let's consider this one:

3+2 The above expression evaluates to five. The above expression has three elements in it: • • •

The number 3 The addition operator '+' The number 2

Well, we have already dealt with the idea that 2 and 3 are numbers representing values in an expression. Here, let's talk about the '+'. Officially, the '+' in the above expression is called an operator. It is the addition operator. Operators do not exist alone in expressions. That is, for example, the addition operator needs some values to add. It needs some values to operate upon. In this expression:

3+2 The addition operator works with, or operates upon, the values represented by the numbers 3 and 2. The numbers 3 and 2 are said to be the operands for the '+', that is, the 3 and the 2 are operands for the addition operator. It is very handy to think of the addition operator as 'grabbing' the 3 and the 2 and, through arithmetic, producing a value of five.

Speaking a bit more technically, we would say that the above expression evaluates to five because the addition operator operates on the 3 and the 2, (the number 3 and the number 2 are its operands), to produce a value of five. So, now we know: • •

Mathematics expressions are made up of operators and operands. When an expression is evaluated operators accept operands, (which are values), and produce new values through arithmetic.

Specifically, regarding the addition operator we see: • • •

The symbol for the addition operator is '+'. The addition operator needs two operands, one to its left and one to its right. The operands for the addition operator are values.

Since the addition operator requires two operands, we say that it is a binary operator. Binary means 'consisting of two parts'.

There are other binary operators present in expressions. There is also the subtraction operator, as in:

7-4 Here, the subtraction operator, '-', accepts the 7 and the 4, and using arithmetic it produces a value of three.

So, we now have two examples of binary operators in mathematics expressions: • •

The addition operator, '+' The subtraction operator, '-'

Now, an expression can have several addition and subtraction operators in it. For example:

7-4+2 The operators are going to accept values as operands, but in this case, which operator gets the first turn at accepting these values, the '-' or the '+'? To answer that question, it would be handy to know which operator is 'stronger'. In mathematics, when one operator is 'stronger' than another, we say that it has precedence over the other. That is, it precedes the other in the order of the steps taken necessary to evaluate the expression. As it turns out the addition operator and the subtraction operator have equal precedence, one is not 'stronger' than the other. In cases like this, when two operators are of equal precedence, evaluation proceeds from left to right. So, in this case the subtraction operator goes first since it is left most. In detail, here is how this evaluation unfolds, starting with the original expression:

7-4+2

The '-' operator goes first. It accepts the value of seven represented by the number 7 and the value of four represented by the number 4, does arithmetic called subtraction, and produces a value of three. Since the value of three can be represented by the number 3, this work effectively changes the above expression to look like this:

3+2 Now, the action is handed over to the '+' operator. It accepts the value of three, (which was handed to it by the '-' operator), and the value of two represented by the number 2, does arithmetic called addition, and produces a value of five. The value of five can be represented by the number 5, so the above expression boils down to this one:

5

At this point we know: • • • • •

In math expressions there are operators and operands. It is the interaction among operators and operands that is called the evaluation of the expression. The evaluation of a mathematics expression produces a final value. Binary operators require two operands. The addition operator, '+', and the subtraction operator, '-', are both binary operators.

• •

The addition operator and the subtraction operator have equal precedence. When two operators have equal precedence, the one to the left accepts operands first, producing a value which is passed over to the other operator.

More About Operators and Operands Now, let's consider this situation:

2*3 The asterisk, '*', in the above equation is meant to show multiplication. So, the above reads 'two times three'. Therefore, in this example the operation of multiplication is shown by '*'. And we say that the symbol '*' is the multiplication operator. There are other ways to show multiplication. An 'X' is often used, and sometimes a dot is used, and sometimes just placing two variables next to each other means multiplication. But for now, let's just use an asterisk to designate multiplication. Notice that multiplication is a binary operation. That is, the multiplication operator, '*', is a binary operator. It accepts two operands. In the above example the two operands are the number 2 and the number 3. The multiplication operator above accepts the two operands, performs multiplication arithmetic with them, and produced the value six. The value six can be represented by the number 6, so, we can write:

6=2*3

The division operator, '/', is also a binary operator. Consider this expression:

10 / 5 This would read 'ten divided by five'. The division operator here accepts two operands, the number 10 and the number 5, and divides the left one by the right one. This produces a value of two. The value of two can be represented by the number 2, so we can write:

2 = 10 / 5 Division and multiplication have equal precedence. Consider this expression:

10 / 5 * 3 As with the similar case involving addition and subtraction, here we proceed from left to right when operators have identical precedence. So, the ten is divided by five. This produces a value of two. This value of two is next multiplied by the value of three to produce a value of six. The value of six can be represented by the number 6, so we have:

6 = 10 / 5 * 3

Here's what we have so far: • • •

Multiplication and division are both binary operators. Multiplication and division have equal precedence. Again, when operators have equal precedence, evaluation happens from left to right.

Now, understand that multiplication and division have precedence over addition and subtraction. Consider this expression:

3+2*4 The multiplication operator, '*', accepts operands, the numbers 2 and 4, before the addition operator, '+'. Therefore, two times four produces a value of eight; three added to eight produces a final value of eleven. This evaluation can be viewed this way:

3+2*4 3+8 11 So, we can write:

11 = 3 + 2 * 4 Here are some more examples to show how multiplication, division, addition, and subtraction interact. Consider:

3+8/4-1 Here, division has precedence over addition and subtraction. So, it accepts operands first. Eight divided by four is two, effectively changing the expression to this:

3+2-1 Now, addition and subtraction have equal precedence. Addition is to the left, so it accepts operands first. Three added to two equals five. So, effectively we now have:

5-1 Finally, the subtraction operator accepts operators. Five take away one is four. So, we can say:

4=3+8/4-1

Also consider:

2 * 4 - 12 / 3 In this expression both the multiplication operator, '*', and the division operator, '/', have precedence over the subtraction operator, '-'. Both the multiplication operator and the division operator have equal precedence, so we proceed from left to right with them. That makes the

multiplication operator accept operands first. Two multiplied by four is eight. So, we effectively have:

8 - 12 / 3 Next in line is the division operator. Twelve divided by three is four, effectively giving us:

8-4 And, of course, finally, eight take away four is four. So, we say:

4 = 2 * 4 - 12 / 3

In summary, so far: • • • •

Binary operators take two operands. Addition, subtraction, multiplication, and division are all binary operations. Multiplication and division have precedence over addition and subtraction. When two operators have equal precedence, the one to the left accepts operands first.

Unary Operators Not all operators in mathematics expressions are binary operators. Some are unary operators. As you may guess, a unary operator accepts only one operand. The positive sign and the negative sign are each unary operators. They accept only one value when they do arithmetic. The positive sign looks a lot like the addition operator. For example, consider this expression:

+2 Is read ‘positive two’. It does not really mean ‘add two’. What is happening here from an expression evaluation viewpoint is that the value of two represented by the number 2 is accepted as a single operand by the unary operator, ‘+’, and that operation produces a value of positive two. But, of course, the value of two represented by the number 2 is positive to start with, so, this operation is not too exciting, (no pun intended). Obviously, ‘+2’ and ‘2’ each represent the same value. So, one can say that an ‘invisible’ positive sign can be imagined before any number, (or variable or function, for that matter). As mentioned above, the negative, (or opposite), sign is a negative operator. Consider this expression: • 3 Technically, here the negative sign operator accepts a value of three as its operand and produces a value of negative three. Obviously, ‘-3’ and ‘3’ represent different values. There are no invisible negative signs in mathematics expressions. These two unary operations, ‘+’ and ‘-‘, have precedence over all binary operations. To see this, let’s look at a situation which is often termed a ‘double negative’ or stated as ‘two negatives make a positive’. Consider this expression:

3 - -2 In this expression there are two operators that look like this: ‘-‘. They are not the same. The first one, (left most), is the binary subtraction operator. The other, (right before the 2), is the unary negative sign operator. From an expression evaluation viewpoint, things proceed as follows. The negative sign operator has precedence over the subtraction operator, so it accepts operands first. It accepts only one operand, since it is a unary operator. It accepts the value of two and produces a value of negative two. Now, the subtraction operator gets its chance to accept

values. It takes the three and from it subtracts the negative two. This produces five, represented by the number 5, so we can write:

5 = 3 - -2 Now, this expression also equals five:

3+2 Since two expressions both equaling five equal each other we can write:

3 + 2 = 3 - -2 Although the expressions to the left and right of the equal sign above produce the same value, they each have quite a different story going on regarding the interaction of operators and operands. So, the positive sign operator is not the addition operator, and the negative sign operator is not the subtraction operator. But they look so much alike! Or do they? Actually, the positive sign operator is often differentiated from the addition operator by making it a bit smaller and lifting it up a bit, as in: +

2 + +3

That would read ‘positive two added to positive three’. A similar visual difference is often used for the negative sign operator and the subtraction operator. What was written above in the ‘double negative’ discussion:

3 - -2 Is much more clearly written this way:

3 - -2 That makes it very clear that the negative sign is a different operator than the subtraction operator. This last expression would be read ‘three subtract negative two’. There is really no ‘double negative’ here. One operation is binary subtraction, the other is unary negation. The combination of these in like expressions may evaluate to a value equal to a similar looking expression involving an addition operator, and an addition operator may look like a positive sign, but it is really not the best mathematics to explain the fact that:

3 + 2 = 3 - -2 Using terms like ‘double negative’ or ‘two negatives make a positive’. So, is there ever a double negative or do two negatives ever make a positive?

Well, yes, if there really are two negative sign operators present, (and not one subtraction operator and one negative sign operator). Consider this expression: • 2 Here there are two unary negative sign operators looking around, as we might say, for operands. They have equal precedence, so, which one accepts operands first? Unlike the situation for binary operators where in situations of equal precedence evaluation proceeds from left to right, here, with unary operators of equal precedence evaluation goes from right to left. So, the value of two is accepted by the right most negative sign producing a value of negative two, or the opposite of two. This value is handed over to the left most negative sign producing the negative of negative two, or the opposite of the opposite of two, which is positive two. In this case, then, it is fine to talk about double negatives and say that two negatives produce a positive. Now, with complete clarity we can write: +

2 = --2

Collecting a few ideas about expression evaluation: • • • •

There are binary and unary operators. Unary operators have precedence over binary operators. Resolutions regarding binary operations of equal precedence proceed left to right. Resolutions regarding unary operations of equal precedence proceed right to left.

Raise to a Power Our final operator has the highest precedence, is binary, and is usually invisible. Here, we are going to discuss the raise to a power operation. Basically, a raise to a power operation looks like this:

23 The two is called the base. The 3 is called the exponent. The whole expression, that is, 23, is called the power. This power would be called ‘the third power of two’. It is often called ‘two raised to the third power’, also. Just remember that, technically, the 3 is not the power. It’s the exponent. Again, the power is 23. This power, 23, evaluates to eight because 23 means two times itself three times, that is, two times two times two. There is no visible operator. It is the positioning of the exponent, the 3 in this example, to the right and up from the base, the 2 in this example, that designates the operation.

This raise to a power operation has precedence over all the binary operators (multiplication, division, addition, subtraction) and unary operators (positive and negative signs). For example, consider this example:

4 * 23 We have two operations present here: multiplication and raise to a power. The raise to a power operation goes first. So, two raised to the third power evaluates to eight, and then four times eight is thirty-two. Therefore, we can write:

32 = 4 * 23 Here is a harder one to understand: • 42 Now, the negative sign out front must wait till the raise to a power operation is finished. So, four raised to the second power is sixteen, since four times four is sixteen. After that

evaluation the negative sign accepts the value of sixteen as an operand and produces a value of negative sixteen. Therefore, we can write: • 16 = -42 In other words, -42 does not mean negative four times negative four. That would be positive sixteen. Some calculators give this result; so, be careful and make sure that you understand how the calculator that you are using works. The exponent may be negative. Consider this expression:

4-3 The negative sign on the exponent means that this power has the value of one divided by four raised to the third power. That is:

1 / 43 = 4-3 Although the above notation is not incorrect in any way, perhaps this is more clear:

The exponent can be a fraction. Taking a root, such as a square root or a cube root, is actually the raising of a number to a fractional power. Here are some examples:

Now, if the exponent contains some arithmetic, all of that arithmetic must be done before you can clearly see with what power you are working. In other words, the exponent itself can be an expression with operators and operands. For example, consider this:

32 + 4 In the above example the exponent is the expression ‘2 + 4’, which evaluates to six. So the entire expression, 32 + 4, evaluates to the sixth power of three, or equivalently three times three times three times three times three times three, or seven hundred and twenty nine. So:

32 + 4 = 36 = 729 Here is an interesting situation:

What does that mean? Well, it means two raised to some power. But what power? Well, the exponent for two contains some arithmetic which itself contains a raise to the power operation. The exponent for two is the fourth power of three, or eighty-one. In notation that looks this way:

By the way, the eighty-first power of two is quite a large number. My calculator reads: 2,417,851,639,229,258,349,412,352. I would suspect that is correct, but I really have no common experience to check it against. Sometimes an operator is shown for the raise to the power operation. This is regularity true when such an expression must be entered into a device that does not permit superscripts, such as the graphics calculator like EZ Graph. Under such conditions a caret, or ‘^’, is used. In other words, this:

3^4 Stands for this: 34

Parentheses Basically, parentheses are used in expressions to change the natural order of precedence among the operators. For example, consider this expression:

3+4*5 Since multiplication has precedence over addition, we know that this expression will be evaluated as follows: Four times five is twenty; twenty plus three is twenty-three. Or, in notation: 3+4*5 3 + 20 23 But what if you want to add the three and the four first, and then multiply by five? How do you then set up the notation? You use parentheses, like this:

(3 + 4) * 5 One always evaluates the expression inside of parentheses first. So, for the above situation three is added to four to get seven, and then seven is multiplied with five to get thirty-five. Again, in notation: (3 + 4) * 5 7*5 35 The expression inside of the parentheses may have its own precedence for operations. For example:

(2 + 4 * 6) * 3 Would work this way: Inside the parentheses multiplication has precedence over addition, so four would be multiplied by six to get twenty-four, and then two would be added to twenty-four to get twenty-six. That is the value inside of the parentheses. This value, twenty-six would then be multiplied by three to finally get a value of seventy-eight. In notation: (2 + 4 * 6) * 3 (2 + 24) * 3 26 * 3 78 Notice, regarding an example which we have seen before: • 42 = -16 That is, the negative of the second power of four is negative sixteen. But by using parentheses we can state:

(-4)2 = 16 That is, the second power of negative four is positive sixteen, or negative four raised to the second power is positive sixteen. Here is another example from before:

Notice how parentheses changes things; here, it is the third power of two which receives its value first, making a value of eight, and then this value is raised to the fourth power:

As we saw before, 281 = 2,417,851,639,229,258,349,412,352. But 84 = 4,096. Certainly changing the natural order of the operators with parentheses changes quite a bit here. Sometimes parentheses have a use that does not look much like they are being used to change the order of operator precedence. They can be used to group terms even if this grouping does not change the order of evaluation. For example, the parentheses are not really needed here, since multiplication automatically has precedence over addition: 4 + (5 * 3) However, they do no harm, and, perhaps, the author wants to show some special feature in a problem where the five times the three forms a unit of thought currently under discussion, so, it is emphasized this way.

Including Variables and Functions If you have been following the links from top to bottom on the Expression Evaluation page, you probably noticed that variables and functions were mentioned early on, and then through most of the material only numbers were used in the examples. That is because the main topics under discussion when explaining the evaluation of expressions concern operators, operands, and precedence. The operands we used were numbers, but any symbol representing a value could have been used, and all of the conclusions would remain the same. In other words, here is an expression that has in it he numbers 4, 5, and 6, and the subtraction and division operators:

4-5/6 As stated earlier, division has precedence over subtraction, so five is divided by six, and this value is then subtracted from four. Here is a similar situation with variables and functions, rather than numbers, representing values:

x - f(x) / y The same rules of precedence apply. The output value of the function, f(x), is divided by the value of y, producing some intermediate value. This intermediate value is then subtracted from x to get the final value for the expression. Of course, you would need to know the actual values for x, f(x), and y to come up with the actual final value for this expression. The purpose here, though, is to note that the order of operations works the same for numbers, variables, and functions.

And now we will examine an interesting note about the raise to a power operation, and how it is written for numbers, variables, and functions. The proper notation for the power of a number, say the second power of 4, is:

42 The proper notation for the power of a variable, say the second power of x, is:

x2 But, the proper notation for the power of a function, say the second power of f(x), is not:

f(x)2 Correctly written, the second power of f(x), or f(x) squared, is:

f2(x) Although it is usually not seen, this would be the same as (f(x))2. So:

f2(x) = (f(x))2 Now, also note that:

4-1 means 1/41 or 1/4 And:

x-1 means 1/x1 or 1/x But:

f-1(x) does not mean 1/f(x)

The symbol f-1(x) means the inverse of the function f(x). And inverse functions really have nothing to do, (necessarily), with reciprocals. If you want to state 1/f(x) using notation similar to above, then:

(f(x))-1 means 1/f(x)

A Few Notes About Multiplication We have used an asterisk for the multiplication operator. This is not really unusual; many computer languages and graphics calculators work this way. So, a statement such as this can mean three times two:

3*2 But there are other common ways to note multiplication. Each of the following are usually accepted to mean three times two:

If you were using variables, such as p and q, multiplication could be shown in any of these ways:

In fact, in such cases involving variables you can indicate a multiplication by simply placing the variables next to each other. The following means p times q: pq The above method does not work for numbers, though. If you try to state three times two using this method you end up with:

32 Which, obviously, is thirty-two, not at all equal to six. If you want to put two numbers next to each other in order to indicate multiplication, you need to use parentheses. Any of the following correctly states three times two: 3(2) (3)2 (3)(2) And what if you want to show a number times a variable? Well, here is five times x:

5x The number comes first, and no parentheses are required. Although strictly not incorrect, the following forms are not normally used in mathematics notation: x5 (5)x 5(x) (5)(x)

Terms and Factors When discussion expression evaluation it is handy to understand what is meant in mathematics by the words ‘term’ and ‘factor’. We have looked at expressions and discussed them from the viewpoint of operators and operands. Now, we will look at an expression from the viewpoint of terms and factors. From this viewpoint we are less concerned with the interactions among the parts of the expression, but more concerned with how to generally name the parts of an expression. Terms are those parts of the expression between addition signs and subtraction signs. Consider this expression: x+y-z This expression has three terms: the variables x, y, and z. We say that the first term is x, that the second term is y, and so on.

Here is another expression:

7 - f(x) + a This expression also has three terms: the number 7, the function f(x), and the variable a. The first term is the number 7, the second term is the function f(x), and the third term is the variable a. Now, consider this expression: xy + 2 Here, the first term is this: xy Notice that this term is made up of two parts, the variable x and the variable y.

Factors are the separate parts of a multiplication. Consider this expression: xy The above, of course, is a multiplication of x times y. The factors in this multiplication are x and y. In the following expression the factors are 8, p, and q:

8pq

Now, back to this expression: xy + 2 Here the first term is xy, and that first term is made up of two factors, x and y. So, terms are groups of factors. The second term in this expression is the number 2. This term is made up of one factor, 2. Consider this expression: cd2 This expression has one term which is made up of two factors. The first factor is c. The second factor is d2. In other words, powers are considered single factors. The factor d2 is not considered two factors even though it is equal to d times d. What about division? Well, it can defined in terms of multiplication, as in:

So, one could say that the factors of a/b are a and 1/b.

And clearly see that the factors of a/b are a and b-1. So, terms are inside of factors, but the factors themselves can be made up of terms which can have factors inside of them, and so it. It can get quite complicated! Consider this: (a + b + c)(d - ef) From the outer most viewpoint, this is an expression with one term, the entire expression. From this outer most viewpoint the first and only term in this expression is made up of two factors. The first factor is: (a + b + c) The second factor of this first and only term is: (d - ef) Now, let’s look at that first factor of the first term. Again it is: (a + b + c) Or simply: a+b+c This factor is itself made up of three terms: a, b, and c. Each of these terms has but one factor since each is but one variable. Let’s also look at the second factor of the first term. Again it is: (d - ef) Or simply: d - ef This factor is made up of two terms, d and ef. The first term here, d, is made up of one factor, the single variable d. The second term here is made up of two factors, e and f. So, considering the original expression: (a + b + c)(d - ef) One could correctly say that the second term of the second factor is ef. Taking it just a bit further, the second factor of the second term of the second factor of the original expression is f.

Linear Functions Linear functions are functions that have x as the input variable, and x is raised only to the first power. Such functions look like the ones in the above graphic. Notice that x is raised to the power of 1 in each equation. Functions such as these yield graphs that are straight lines, and, thus, the name linear. Linear functions come in three main forms. Slope-Intercept Form Point-Slope Form General Form Equation of a Line - Slope-Intercept Form y = mx + b Above is a program that will help you visualize how changing the values for the slope, m, and the y-intercept, b, will affect the graph of the equation y = mx + b. At first the program will be automatically cycling through several values for m and b. If you want to use the sliders to control it yourself, just press the ‘You Control’ button. Notice that when the slope, m, is positive, the line slants upward to the right. The more positive m is, the steeper the line will slant upward to the right. When the slope is negative, the line slants downward to the right, and, as the slope becomes more and more negative, the line will slant downward steeper and steeper to the right. Also, notice that when the y-intercept, b, is positive, the line crosses the y-axes above y = 0. When b is negative, the line crosses the y-axis somewhere below y = 0. In fact, b is the value on the y-axis where the line passes through this axis. The line intercepts, or crosses, the y-axis here, and, therefore, b is called the y-intercept.

Summary of Details This linear function:

f(x) = mx + b May be graphed on the x, y plane as this equation: y = mx + b • This equation is called the slope-intercept form for a line. • The graph of this equation is a straight line. • The slope of the line is m. • The line crosses the y-axis at b. • The point where the line crosses the y-axis is called the y-intercept. • The x, y coordinates for the y-intercept are (0, b). Equation of a Line - Point-Slope Form y = m(x - x 1 ) + y 1

or

y - y 1 = m(x - x 1 )

Above is a program that will help you visualize how changing the values for the point, (x1, y1), and for the slope, m, will affect the graph of the equation y = m(x - x1) + y1. At first the program will be automatically cycling through several values for x1, y1, and m. If you want to use the sliders to control the program yourself, just press the ‘You Control’ button. Notice the point, shown as a little square, which moves around the graph. That point is (x1, y1). The point-slope form equation gets its name from the fact that if you know one point on the line, (x1, y1), and the slope of the line, m, then you can determine, or draw, the line completely. Experiment with controlling the program yourself. You should be able to realize that any line which you can imagine on the x, y plane can be drawn knowing only one point on the line and the slope of the line.

Summary of Details This linear function: f(x) = m(x - x1) + y1 May be graphed on the x, y plane as this equation: y = m(x - x1) + y1 This equation is often also written as: y - y1 = m(x - x1) • This equation is called the point-slope form for a line. • The graph of this equation is a straight line. • A known point on the line is (x1, y1). • The slope of the line is m. Equation of a Line - General Form

Ax + By + C = 0

or

y = (-A/B)x + (-C/B)

Above is a program that will help you visualize how changing the values for the variables A, B, and C will affect the graph of the equation y = (-A/B)x + (-C/B). At first the program will be automatically cycling through several values for A, B, and C. If you want to use the sliders to control it yourself, just press the ‘You Control’ button. Notice that the slope is equal to the opposite of A divided by B, that is, (-A/B). Also, notice that the y-intercept is equal to the opposite of C divided by B, that is, (-C/B).

Summary of Details This linear function: f(x) = (-A/B)x + (-C/B) May be graphed on the x, y plane as this equation: y = (-A/B)x + (-C/B) This equation is often also written as:

Ax + By + C = 0 • • • •

This equation is called the general form for a line. The graph of this equation is a straight line. The slope of the line is (-A/B). The y-intercept is (-C/B).

Polynomial Functions Stated quite simply, polynomial functions are functions with x as an input variable, made up of several terms, each term is made up of two factors, the first being a real number coefficient, and the second being x raised to some non-negative integer power. Actually, it’s a bit more complicated than that. Definition of a Polynomial Function Here a few examples of polynomial functions: f(x) = 4x3 + 8x2 + 2x + 3 g(x) = 2.5x5 + 5.2x2 + 7 h(x) = 3x2 i(x) = 22.6 Polynomial functions are functions that have this form: f(x) = anxn + an-1xn-1 + ... + a1x + a0 The value of n must be an nonnegative integer. That is, it must be whole wsnumber; it is equal to zero or a positive integer. The coefficients, as they are called, are an, an-1, ..., a1, a0. These are real numbers. The degree of the polynomial function is the highest value for n where an is not equal to 0. So, the degree of g(x) = 2.5x5 + 5.2x2 + 7 is 5. Notice that the second to the last term in this form actually has x raised to an exponent of 1, as in: f(x) = anxn + an-1xn-1 + ... + a1x1 + a0 Of course, usually we do not show exponents of 1. Notice that the last term in this form actually has x raised to an exponent of 0, as in:

f(x) = anxn + an-1xn-1 + ... + a1x + a0x0 Of course, x raised to a power of 0 makes it equal to 1, and we usually do not show multiplications by 1. So, in its most formal presentation, one could show the form of a polynomial function as: f(x) = anxn + an-1xn-1 + ... + a1x1 + a0x0 Here are some polynomial functions; notice that the coefficients can be positive or negative real numbers. f(x) = 2.4x5 + 1.7x2 - 5.6x + 8.1 f(x) = 4x3 + 5.6x f(x) = 3.7x3 - 9.2x2 + 0.1x - 5.2 Roots of Polynomial Functions

Real roots Synthetic Division Real Roots of Polynomial Functions Definition of terms and symbols when dividing polynomials:

Dividend:

f(x)

Divisor:

h(x)

Quotient:

q(x)

Remainder: r(x) If any of these are constants, for example if r(x) is constant, as in:

r(x) = 5 or: r(x) = a then variable, rather than function, notation may be used for that value, as in: r=5 or: r=a When f(x) is divided by h(x), the result is the value of q(x) plus r(x), as in: f(x)/h(x) = q(x) + r(x) This can also be written as: f(x) = h(x)q(x) + r(x)

The remainder, r(x), will either be equal to 0, or it will be less in degree than the degree of the divisor, h(x). If h(x) has a degree of 1, then the degree of the remainder must be 0. That is, the remainder must be a constant, as in: r(x) = cx0 = c Under these conditions variable notation is fine, as in: r=c Therefore, if f(x) is divided by the linear polynomial (x - c), the remainder is a constant, r. Again, consider our basic definition of polynomial division: Dividend: f(x) Divisor: h(x) Quotient: q(x) Remainder: r(x) f(x) = h(x)q(x) + r(x) Make the divisor, h(x), equal to the zero degree polynomial (x - c). This will create a remainder, r, that is a constant. h(x) = (x - c) Then this: f(x) = h(x)q(x) + r(x) Becomes: f(x) = (x - c)q(x) + r Let x = c, so: f© = (c - c)q(x) + r f© = (0)q(x) + r f© = 0 + r f© = r Therefore, when the polynomial function, f(x), is divided by a linear polynomial function in the form (x - c), the remainder is f©. This we will call the remainder theorem for polynomial division. Using this remainder theorem, if the divisor is the linear function (x - c) as in: h(x) = (x - c) Then our basic definition of polynomial division: f(x) = h(x)q(x) + r(x) Becomes: f(x) = (x - c)q(x) + f©

Suppose c is a zero of f(x). Then, of course: f© = 0 And the above: f(x) = (x - c)q(x) + f© Would become: f(x) = (x - c)q(x) + 0 Or: f(x) = (x - c)q(x) That clearly makes (x - c) a factor of f(x). Therefore, if c is a zero of f(x), then (x - c) is a factor of f(x). That means that f(x) can be though of as the product of (x - c) times some other polynomial function, q(x). This could be summarized as: If f© = 0 Then f(x) = (x - c)q(x) So, whenever we know a root, or zero, of a function, we know a factor of that function. Now we are in a position to understand a method for analytically solving a certain group of problems regarding finding roots of polynomial functions. Suppose you have a polynomial function of degree 3, and you wish to find the real, possibly integer, roots. This function might look like: f(x) = x3 - 9x2 + 26x - 24 Now, suppose you knew one root for this function. One happens to be x = 2. Check: f(2) = 23 - 9(22) + 26(2) - 24 f(2) = 8 - 36 + 52 - 24 f(2) = 0 That means: f(x) = (x - 2)q(x) Or: x3 - 9x2 + 26x - 24 = (x - 2)q(x) We can find q(x) by dividing f(x) by (x - 2), as in: q(x) = (x3 - 9x2 + 26x - 24) / (x - 2) This yields: q(x) = x2 - 7x + 12 Which means that this:

f(x) = (x - 2)q(x) Becomes: f(x) = (x - 2)(x2 - 7x + 12) Now, the second factor, (x2 - 7x + 12), is easily factored, as: (x2 - 7x + 12) = (x - 3)(x - 4) So, this: f(x) = (x - 2)(x2 - 7x + 12) Becomes: f(x) = (x - 2)(x - 3)(x - 4) Therefore we have factored f(x) into a group of linear factors. The roots of f(x) are clearly demonstrated. They are 2, 3, and 4. So, if we know one root of a cubic polynomial function, we know a linear factor of that function. If we know a linear factor of the cubic we can divide the cubic by that factor and get another polynomial factor of one degree less than the cubic. That is, we will get a quadratic polynomial function as a quotient. That quadratic can be factored by convention means, certainly by the quadratic equation. This factored quadratic equation will yield the other linear factors of the original cubic. So, a cubic can be factored into a group of linear factors using these methods. When it is so factored, its roots are obvious. This method can be generalized to situations concerning finding the roots of polynomial functions with degrees past the third. For example, if you had a polynomial function of the fourth degree, and if you could find one zero of that function, then you would have one linear factor of that function. You could divide the fourth degree polynomial by this linear function and find a polynomial of degree three. This third degree polynomial function could be factored as described above. At the moment there are some things to be concerned with: The above methods depend upon knowing at least one zero of the polynomial function so as to generate that first linear factor which will subsequently be used as a divisor to reduce the the degree of the original polynomial function. How does one find this zero? Once the process has gone on long enough, and enough linear functions have been found and used as divisors until the current quotient is of degree two and can easily be factored using the quadratic equation, what does it mean if this quadratic function factors into less than two linear factors, say one or none?

Synthetic Division Consider this polynomial function:

f(x) = 4x3 - 3x2 + x - 4 Suppose that we evaluate it at an input of x = 2, like this: f(2) = 4(23) - 3(22) + (2) - 4 f(2) = 32 - 12 = 2 - 4 f(2) = 18 In this process we raised the input to a power, as in 23. Let us see that there is a way to evaluate this polynomial function using only multiplication and addition. Start with the original polynomial and factor out an x. So, this: 4x3 - 3x2 + x - 4 Becomes: x(4x2 - 3x + 1) - 4 Factor out another x from the parenthesized expression: x(x(4x - 3) + 1) - 4 Now, imagine that you evaluate f(x) at x = 2. Begin with the inner most expression. Place a 2 for the input value of x, as in: x(x(4(2) - 3) + 1) - 4 Now you would multiply 2 (the input) by 4 (the original coefficient of x3) and then add -3 (the original coefficient of x2). This would evaluate to 5. The expression now looks like: x(x(5) + 1) - 4 Place a 2 for the next input value of x, as in: x(2(5) + 1) - 4 Now you would multiply 2 (the input) by 5 and then add 1 (the original coefficient of x). This would evaluate to 11. The expression now looks like: x(11) - 4 Place a 2 for the last input value of x, as in:

2(11) - 4 Now you would multiply 2 (the input) by 11 and then add -4 (the original final constant of the polynomial). This would evaluate to 18, the same value which we calculated at first, of course. Now, this method of evaluating the output value of a polynomial can be captured in a type of short-hand notation by positioning the input value and the coefficients of the polynomial in a certain form. Again, consider this polynomial: f(x) = 4x3 - 3x2 + x - 4 We will picture it evaluated at the input value x = 2. Arrange the input value, the coefficients, and a line like this: 2) 4 -3 1 -4

Now drop down the 4: 2) 4 -3 1 -4 4 Multiply the input 2 times the 4. Place this product, 8, under the -3:

2) 4 -3 1 -4 8 4 Add the -3 and the 8. Place this sum, 5, under the line:

2) 4 -3 1 -4 8 4 5 Multiply the input 2 times the 5. Place this product, 10, under the 1:

2) 4 -3 1 -4 8 10 4 5 Add the 1 and the 10. Place this sum, 11, under the line:

2) 4 -3 1 -4 8 10 4 5 11 Multiply the input 2 times the 11. Place this product, 22, under the -4:

2) 4 -3 1 -4 8 10 22 4 5 11 Add the -4 and the 22. Place this sum, 18, under the line:

2) 4 -3 1 -4 8 10 22 4 5 11 18

We are done. This final value, 18, is the function output for f(x) evaluated with an input of 2. This process is called synthetic division. At first this term may be confusing. What does a method of function evaluation have to do with division? Consider dividing f(x) = 4x3 - 3x2 + x - 4 by (x - 2). Standard polynomial division would look like this: 4x2 + 5x + 11

--------------------3

x - 2 )4x - 3x2 + x - 4

4x3 - 8x2 5x2 + x 5x2 - 10x 11x - 4 11x - 22 18 Compare the above with our previous synthetic division:

2) 4 -3 1 -4 8 10 22 4 5 11 18 Notice that the remainder from the standard polynomial long division is the last value calculated in the synthetic division. Notice that the coefficients of the quotient from the standard polynomial long division are aligned before the remainder on the last row, under the line, of the synthetic division. So, this synthetic division method can be used to determine the quotient and remainder when a polynomial function, f(x), is divided by a linear function of the form (x - c). Specifically, this method can be used to determine if a certain value, c, is a root of the polynomial function f(x). If c is a root, then the synthetic division method will show a zero in the remainder position. This would be equivalent to determining that (x - c) is a linear factor of f(x). An added benefit of this method is that it yields the coefficients of the quotient when f(x) is divided by (x - c). This quotient can be further examined and factored using synthetic division until all the linear factors of the original polynomial function, f(x), have been found. Once all the linear factors of the polynomial function have been found, all the roots of the function are obvious.

Polynomial Function Graphs The following are Java applets that demonstrate and animate the graphs of several polynomial functions. Zero Degree First Degree Second Degree Third Degree Fourth Degree Fifth Degree

Polynomial Function of Zero Degree A zero degree polynomial function can be defined like this:

f(x) = a 0th Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficient a. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. Usually, a zero degree polynomial function is called a constant function. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0 Polynomial Function of the First Degree

A first degree polynomial function can be defined like this:

f(x) = ax + b 1st Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a and b. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. Usually, a first degree polynomial function is called a linear function. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0

Polynomial Function of the Second Degree A second degree polynomial function can be defined like this:

f(x) = ax2 + bx + c 2nd Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a, b, and c. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. Usually, a second degree polynomial function is called a quadratic function. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0 Polynomial Function of the Third Degree

A third degree polynomial function can be defined like this:

f(x) = ax3 + bx2 + cx + d 3rd Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a, b, c, and d. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0 Polynomial Function of the Fourth Degree A fourth degree polynomial function can be defined like this:

f(x) = ax4 + bx3 + cx2 + dx + e 4th Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a, b, c, d, and e. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0 Polynomial Function of the Fifth Degree

A fifth degree polynomial function can be defined like this:

f(x) = ax5 + bx4 + cx3 + dx2 + ex + f 5th Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a, b, c, d, e, and f. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0

Rational Functions A rational function is formed when a polynomial is divided by another polynomial. Below are some links to sections dealing with rational functions. There is a wealth of material here that would be helpful to any student of mathematics studying this topic.

The following link takes you to an overview of rational functions. It presents a definition of a rational function as long as some introductory material concerned with finding the roots and discontinuities of rational functions.

Rational Function Definition These next links present some fundamental ideas concerning one of the simplest of rational functions, f(x) = 1/x.

A Very Simple Rational Function: f(x) = 1/x Transformations of f(x) = 1/x This next link gives a detailed explanation of how to work with a rational function. This includes a complete presentation of how to find roots, discontinuities, and end behavior.

Simple 2nd Degree / 2nd Degree The following links are all to special purpose graphing applets that each present a common rational function. These are useful to expand your understanding of the material covered in the above material. 1st Degree / 1st Degree 1st Degree / 2nd Degree 1st Degree / 3rd Degree 2nd Degree / 1st Degree 2nd Degree / 2nd Degree 2nd Degree / 3rd Degree 3rd Degree / 1st Degree

3rd Degree / 2nd Degree 3rd Degree / 3rd Degree Definition of a Rational Function A rational function is basically a division of two polynomial functions. That is, it is a polynomial divided by another polynomial. In formal notation, a rational function would be symbolized like this:

Where s(x) and t(x) are polynomial functions, and t(x) can not equal zero.

An Example Here is an example of a rational function:

To understand the behavior of a rational function it is very useful to see its polynomials in factored form. The polynomials in the numerator and the denominator of the above function would factor like this:

The Domain Now the roots of the denominator are obviously x = -3 and x = 6. That is, if x takes on either of these two values, the denominator becomes equal to zero. Since one can not divide by zero, the function is not defined for these two values of x. We say that the function is discontinuous at x = -3 and x = 6. Other values for x do not cause the function to become undefined, so, we say that the function is continuous at all other values for x. In other words, all real numbers except -3 and 6 are allowed as inputs to this function. The domain for the function, therefore, as expressed in interval notation is:

The x-intercepts The x-intercepts for this function would be where the output, or y-value, equals zero. A rational function can be considered a fraction, and a fraction is equal to zero when the numerator is equal to zero. For our rational function example this happens when the polynomial in the numerator is equal to zero, and this will happen at the roots of this polynomial. The roots of the numerator polynomial are x = -5 and x = 4. That is, when x takes on either of these two values the numerator becomes zero, and the output of the function, or y-value, also becomes zero. So, the x-intercepts for this rational function are x = -5 and x = 4. Notice that the function is defined at these two values. (It is only not defined at x = -3 and x = 6.) That makes these true x-intercepts. If the function was not defined at x = 4 because 4 was a root of the denominator polynomial, (which it is not in our example here), then x = 4 would not be an x-intercept even though it made the numerator equal to zero. One can not have an xintercept for a function at a point where the function does not exist!

The y-intercept What about the y-intercept? Well, they occur where the input, or x, value equals zero. If we look at our first un-factored form for this function, expressed in ‘y =’ form, we have:

Now, setting x = 0 we get:

That is, the graph crosses the y-axis at y = 10/9 (about 1.11). Notice that when you express the polynomials of a rational function in standard form, then the y-intercept is simply the ratio of the final terms for the two polynomials.

The Graph Let us look at a graph of this function:

The red lines are the x- and y-axes; the blue grid marks off unit distances; the green line is the function. Realize the following: •

• •

The function line is discontinuous, or ‘breaks’, at x = -3 and x = 6. That is, if you were drawing the graph by hand, you would have to lift the pen off the paper at at x = -3 and x = 6. That is what we mean by a discontinuity. Notice that these locations for the discontinuities, (x = -3 and x = 6), are the same as we reasoned above. That is, x = -3 and x = 6 are the roots of the polynomial in the denominator. The function crosses the x-axis at x = -5 and x = 4. These are the same as the values which we calculated above for the x-intercepts. The function crosses the y-axis just a bit above 1, at about 1.1. This is the same location as the calculated y-intercept above.

The End Behavior Model Function The way that a function is shaped when the input, or x, is very large positively or very large negatively is called its end behavior. That is, the end behavior of a function describes how it acts as x approaches positive and negative infinity. One might say that the end behavior of a function is described by how it ‘heads out’ to the right and left on the graph far away from the origin. In the graph above notice how the graph is leveling off to a height of about y = 1 as x gets large positively or negatively. That is, at the left and right edge of the graph the function starts to act like the function described as:

This function, y = 1, is called the end behavior model function for the rational function on the graph. You can find it without drawing the graph by dividing the leading term of the polynomial in the numerator by the leading term of the polynomial in the denominator. Our function under consideration is this:

The leading term of the polynomial in the numerator is:

The leading term of the polynomial in the denominator is:

The division of these two leading terms, numerator leading term over denominator leading term, is:

And, of course, this makes the following end behavior model function:

The Range An examination of the graph above will indicate that all real numbers are available for output. The left most fork of the graph looks like it will rise not quite to y = 1, and the right most fork looks like it may not quite drop to y = 1; so, one might at first suggest that y = 1 is not in the range. However, if you examine the central portion of the graph, you

can see that y = 1 is definitely a member of the range. So, all real numbers are present in the range. In interval notation the range would look like this:

Summary This has been an introduction to rational functions and their behaviors. For more details and examples, examine the other topics in the rational function section of the Function Institute. The link labeled ‘Simple 2nd Degree / 2nd Degree’ goes over the material presented here in much more detail and contains an animated Java applet rational function graphing utility, for example. f(x) = 1 / x This is probably the simplest of rational functions:

Here is how this function looks on a graph with an x-extent of [-10, 10] and a y-extent of [-10, 10]:

First, notice the x- and y-axes. They are drawn in red. The function, f(x) = 1 / x, is drawn in green. Also, notice the slight flaw in graphing technology which is usually seen when drawing graphs of rational functions with computers or graphic calculators. At the bottom center

of the picture you will see that the graph line appears to be heading toward the edge of the diagram, but is cut short of that. Actually, the true graph of the function continues downward past the edge of the picture. As we will shortly see, this section of the graph holds what is termed an asymptote, and computers, along with graphic calculators, often have a difficult time drawing functions near asymptotes. Notice that for this function a small positive input value yields a large positive output value. And notice that a large positive input value yields a small positive output value. Here is a picture showing that:

A complementary situation occurs for negative values. A small negative input will output a large negative value, and a large negative input will output a small negative value. Here is a picture showing this idea:

This makes complete sense if you think about it for a moment. Consider a large positive input value, say one million, i. e., 1,000,000. The output of f(x) = 1/x would be one millionth, i. e., 1/1,000,000 or 0.000001. This and other representative examples are shown in the following table:

Input value, or x

Output value, or y

1,000,000

0.000001 (or 1/1,000,000)

0.000001 (or 1/1,000,000)

1,000,000

-1,000,000

-0.000001 (or -1/1,000,000)

-0.000001 (or -1/1,000,000) -1,000,000 Let us look at this function as it leaves the graph at the top and bottom. You should notice that the green function line approaches, but does not touch, the y-axis. If you graphed the function on a set of x, y axes that went up to positive one million and down to negative one million, the function line would still not touch the y-axis, though it would get very close. Just think about the x, y values in the table above. At, say, an output value, or y value, of 1,000,000 the input would not be 0. And, of course, the input value, or x value, must be 0 for the graph to touch the y-axis. This type of behavior about the y-axis is called asymptotic behavior. And, in this case, the y-axis would be called a vertical asymptote of the function. That is, the function approaches the y-axis ever closer and closer, but never touches it. Notice that the x-axis functions as a horizontal asymptote for this function. That is, as the function line stretches out to the left or right it gets closer and closer to the x-axis, but it never touches it. So, for the function f(x) = 1/x the y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote. In the following diagram of this function the asymptotes are drawn as white lines.

The function f(x) = 1/x is an excellent starting point from which to build an understanding of rational functions in general. It is a polynomial divided by a polynomial, although both are quite simple polynomials. Be sure that you understand the concept of an asymptote, especially a vertical asymptote, and then go on to the other rational function information.

Transform of f(x) = 1 / x This Java applet demonstrates several transforms of the function f(x) = 1/x. You can assign different values to a, b, h, and k and watch how these changes affect the shape of the graph.

2nd Degree / 2nd Degree Below is a Java applet which demonstrates the behavior of a rational function. The type of rational function which it demonstrates is a second degree polynomial divided by a second degree polynomial, that is, a quadratic divided by a quadratic. Right below the applet is a very brief discussion of how to use it. That is followed by a detailed discussion of the applet and of rational function behavior in general. This applet graphs this function:

Quick instructions • • • • • • •

The graph has an x-extent of [-10, 10], and a y-extent of [-10, 10]. The x- and yaxes are drawn in red. Variables a and b are the roots of the numerator. Variables c and d are the roots of the denominator. Variables a, b, c, and d are controlled by neighboring scrollbars. Vertical asymptotes are drawn as white lines. Removable discontinuities are drawn as small white circles. Checking “Numerator” causes the polynomial in the numerator to be drawn along with the rational function. Checking “Denominator” causes the polynomial in the denominator to be drawn along with the ration function.

Detailed explanation of applet and rational function behavior in general

Roots Again, this graphing program presents the graphs of rational functions which are second degree polynomials divided by second degree polynomials. An example of such a rational function would be:

You need to visualize this rational function with the polynomials in factored form. That is, the above function, f(x), would need to be seen as:

When investigating rational functions, the first move is usually to factor the polynomials like above. You factor both polynomials to find their roots. In the above example the roots of the numerator are: roots of the numerator: x = 3 and x = -5

And the roots of the denominator are: roots of the denominator: x = 6 and x = -2 In the above applet, the variables a and b are the roots of the numerator, and c and d are the roots of the denominator. So, you should imagine rational functions of this form:

If you were to multiply the above factors in the numerator and denominator, you would get a function that looked like this:

This last rational function might look a bit more normal than its factored version if one is thinking about the division of two polynomials, since polynomials are most often viewed in standard form. However, the roots of the polynomials are at the heart of this discussion, and finding them is the first step in analyzing a rational function.

To understand rational functions it is important to understand the roots of the polynomials that make up the rational function.

Discontinuities Now, one can not divide by zero, so, when the polynomial in the denominator is equal to zero, the rational function is undefined. That is, there may be certain input values for x that cause the polynomial function in the denominator to output a zero. Consider, again, this function:

And, to clearly see the roots, perhaps it is better to view the function like this:

If x = 6, then the value of the denominator is zero since (x - 6) becomes (6 - 6), which is zero, and zero times the other factor, stated as (x + 2) or (x - (-2)), but now valued at (6 + 2) = 8, is zero. That is, zero times 8 is zero. The function is undefined at x = 6 because you can not divide by 0. For similar reasons the function will be undefined at x = -2. This value for x also makes the denominator equal to zero. At the places where the function is undefined we say that the function is discontinuous. And we say that there are discontinuities in the function at the x-coordinates where it is undefined. Therefore, in the above function there are discontinuities at x = 6 and at x = 2.

Discontinuities are locations for x values that make the function undefined.

Asymptotes These discontinuities at x = 6 and at x = -2 show up on the graph as vertical asymptotes. Not all discontinuities in rational functions create vertical asymptotes, but these do. Some discontinuities are not asymptotic, but are removable. Vertical asymptotes show up as vertical white lines on the above applet. To plug the above function into the graphing applet you must set a, b, c, and d to the following values:

a = 3 b = -5 c = 6 d = -2 That is, make the roots of the quadratic in the numerator equal to 3 and -5, and make the roots in the denominator equal to 6 and -2. Go ahead and try that. It should look like the following picture. (Do not check the “Numerator” nor “Denominator” checkbox.)

Notice that the above graph shows a vertical asymptote at x = 6. (Remember, the red lines are the x- and y-axes. Each blue line is an increment of 1.) There is also a vertical asymptote at x = -2, but let us concentrate on the one at x = 6 for the moment. The function is undefined at x = 6 because that input value makes the denominator of the function equal to 0. To the immediate left of 6 the function outputs very large negative numbers making the graph of the function drop very steeply. We could prove this behavior by finding the value of the function with an input value a tiny bit to the left of x = 6. Moving over by 0.001 usually works fine. So, let us find the output value for x = 5.999:

Clearly, to the immediate left of the asymptote the function takes on an output value much more negative than can be seen on the above picture. The above graph only goes to -10 along the y-axis. Let us see what is happening to the immediate right of the vertical asymptote at x = 6. Again, we will step over by the tiny step of 0.001. This would create an input value of 6.001. Here is the output calculation:

Well, it certainly looks like the function takes on very large positive values to the immediate right of x = 6. This behavior, large negative output to the left of x = 6 and large positive output to the right of x = 6, is what causes the asymptotic shape of the function at x = 6. It is important to remember that the vertical asymptote is not a line that is part of the graph of the function. For this reason the asymptote is drawn in a different color. It is white, the actual function is green. The white asymptote line is a visual convenience in the diagram to show a line that the function approaches, but never touches. It is a good idea to see the graph of the function without this aid. The only problem with that is that often asymptotic behavior is not completely drawn to the edges of the diagram by computers. Ultimately, this is due to the fact that the computer diagram is made up of tiny discrete dots, called pixels, while the x, y real number plane is continuous. All the same, un-check the “Asymptotes” checkbox and redraw the function. It should look like this:

Discontinuities may be asymptotic.

Discontinuities and the denominator polynomial As we have seen this function has two discontinuities at the roots of the polynomial in the denominator, x = 6 and at x = -2. Now, the denominator is a second degree polynomial, often called a quadratic polynomial. A second degree polynomial graphs as a parabola. So, if we could see the graph of the denominator polynomial, we would see a graph of a parabola. Where that parabola crosses the x-axis locates the roots, or zeroes, of that parabola. These, of course, are the locations of the asymptotic discontinuities which we have been discussing. To see both the denominator parabola and the rational function graphed together check the ‘Denominator’ checkbox. It might be useful to have the asymptotes turned on, also. Try that. Here is a diagram that shows what it should look like, along with an explanatory note:

Discontinuities are located at the roots of the denominator polynomial.

X-intercepts This function crosses the x-axis at two points. These points are called its x-intercepts. Simply put, an x-intercept will exist where the y, or output, value of the function equals zero. Now, the function we consider here is a rational function, so, since it is a division of polynomials, it may be considered a fraction. A fraction will be equal to zero only when its numerator is equal to zero. The numerator will be equal to zero at the roots of the polynomial in the numerator. The roots of the numerator polynomial are x = 3 and x = -5, as another quick look at our rational function in factored form will reveal:

So the x-intercepts should be at x = 3 and x = -5, since these make the y, or output, value of the function equal to zero. Another look at our current graph will demonstrate this:

X-intercepts are where y =0 (and the function is defined).

X-intercepts and the numerator polynomial The numerator polynomial graphs as a parabola, since it is of second degree. The roots of this parabola are the values which make the numerator of the rational function equal to zero, as explained above. On the applet above if you check the ‘Numerator’ checkbox, then the numerator polynomial will be drawn along with the rational function. You should see that the roots of this parabola are the x-intercepts discussed directly above. Try that. Probably best seen with the asymptotes turned off, it should look like the following:

The roots of the numerator polynomial are the xintercepts (if the function is defined there).

Y-intercept The y-intercept of a function occurs at the point where the function line crosses the yaxis. This would be where x = 0. So, if you set x equal to zero in the rational function, the function will output a value equal to the y-intercept. This is actually quite easy to do for rational functions. Since both the numerator and the denominator are polynomials in x, every term except the last constant term, (if present), in each polynomial becomes equal to zero. So, the value for the y-intercept is simply the ratio of the two last constant terms. This is easier to see than to explain. Here is our function again:

Or, expressed in ‘y =’ form:

Now, to find the y-intercept plug in x = 0 and solve:

Again, let us look at our graph. Notice that it crosses the y-axis at y = 1.25:

The y-intercept is the ratio of the two last constant

terms of the numerator and denominator polynomials.

End behavior model function A rational function is said to have an end behavior model function. The behavior of the end behavior model function and the behavior of the rational function are the same at large positive and large negative values of x. In other words, the rational function and the end behavior model function act the same way as they each ‘head out’ to distant left and right values along the x-axis. The end behavior model function, which we will here name m(x), is found by dividing the leading term of the numerator polynomial by the leading term of the denominator polynomial. One, of course, must view the rational function with the polynomials in standard form, not factored form. Here is another look at our function, polynomials in standard form:

For the function considered here the leading term of the numerator is x2, and the leading term of the denominator is also x2. The division of these two, as described above, yields the end behavior model function, m(x), as shown by:

Or, if you want to consider this end behavior model function in ‘y =’ form, it would look like:

This, of course, is a horizontal line that crosses the y-axis at y = 1. That means that our rational function will approach this horizontal line for large positive and large negative input, or x, values. Although the graph on the applet above does not go far enough to the left or right to see this perfectly, one can see this end behavior fairly clearly. Below is our current graph with the end behavior model function drawn in as a white line:

You should notice that for the rational functions which are presented with this applet the only possible end behavior model is m(x) = 1, since the division of the leading terms of the polynomials in the numerator and denominator will always be x2 / x2.

The end behavior model function demonstrates the behavior of the rational function at extreme values for x.

Discontinuities which are not asymptotes Go to the applet and set up the following situation. Be sure to notice that the only difference is that now x = 6 is a root of both the numerator and the denominator:

a = 6 b = -5 c = 6 d = -2 The graph should now look like this:

Notice that there used to be an asymptote at x = 6, but now there is not. (The asymptote at x = -2 is still there. We are not discussing that here.) There is still a discontinuity at x = 6, that is, the function is still not defined at there because this value for x still makes the denominator equal to zero. And still if one were to imagine drawing the function line carefully, one would have to break the function line by picking up one’s pen at x = 6. That break is represented on the graph by the little white circle. However, this other type of discontinuity is not asymptotic, it is termed removable. We say that there is a removable discontinuity at x = 6. The discontinuity at x = 6 is called removable because it can be removed, (or ‘plugged’), simply by redefining the function at this point. That is, we could assign a certain value to the output, any value, really, at x = 6. A piecewise-definition of the function would work fine, such as:

Notice that all we did was use the normal definition of our rational function for all values of x except x = 6, and, of course x = -2, which is our asymptotic discontinuity. Then we gave an output value of 1 to the function when x = 6. To remove the discontinuity it is not necessary to ‘plug’ the hole in the function line. One simply must provide an output value for the function for the value of x that causes the removable discontinuity. There is, of course, an actual (x, y) point on the real number plane that would ‘plug’ the hole, and,

perhaps, in some applications one would want to know it and use it in a similar piecewise-definition for the function. Finding that point is a calculus problem. Lastly, let us notice something about x-intercepts with this version of our rational function. Suppose we did not redefine the function so as to remove the discontinuity at x = 6. Then we are back to the function in this form:

It may look at first glance that there should be an x-intercept at x = 6 because this value makes the numerator equal to zero, which makes the output, or y, value equal to zero, and that is what we mean by an x-intercept. Except the function can not have an x-intercept at x = 6 because it is not defined there. It is discontinuous, (with a removable discontinuity), at x = 6.

Removable (and not asymptotic) discontinuities form when the the numerator and denominator polynomials share common roots. Conclusion

Well, if you got this far, you gone through most of what you need to know to understand rational functions of many types, not just second degree polynomials divided by second degree polynomials. For example, this same reasoning would be used in an investigation of a first degree polynomial divided by a third degree polynomial. Go back to the applet and play with it. Try out several combinations of for roots both in the numerator and denominator. Adjust it till you understand how to generate asymptotic and removable discontinuities. And, of course, go on to explore the other topics about rational functions in Zona Land.

P. S., So why is this called the simple version? Perhaps you remember that the link from the rational functions home page labeled this ‘Simple 2nd Degree / 2nd Degree’. Well, observant mathematicians will notice that this applet will not graph every type of rational function that is a second degree polynomial divided by a second degree polynomial. It only demonstrates those in which the leading coefficient is equal to one in both the numerator and the denominator. For example, you could not study this rational function with this applet:

This function factors like so:

The applet here provides no way to enter the factor of 3 in the numerator nor the factor of 4 in the denominator. However, the behavior of such functions as this one can be understood using the same methods described here. In fact, the behavior of most rational functions can be understood with these methods. As one closing note, see that the end behavior model function for the rational function above is not y = 1, but:

The end behavior model function is still a horizontal line. This is really the only difference one would need to consider in understanding this type of second degree polynomial over second degree polynomial rational function. Rational functions of this more general type will be demonstrated in other obvious sections in the Rational Functions Department of the Function Institute.

1st Degree / 1st Degree Here’s a grapher that shows the behavior of a rational function made up of a first degree polynomial divided by a first degree polynomial. These polynomials are presented in factored form in the following function definition:

Root of numerator: a. Root of denominator: d

1st Degree / 2nd Degree Here’s a grapher that shows the behavior of a rational function made up of a first degree polynomial divided by a second degree polynomial. These polynomials are presented in factored form in the following function definition:

Root of numerator: a Roots of denominator: d, e 1st Degree / 3rd Degree Here’s a grapher that shows the behavior of a rational function made up of a first degree polynomial divided by a third degree polynomial. These polynomials are presented in factored form in the following function definition:

Root of numerator: a Roots of denominator: d, e, f 2nd Degree / 1st Degree Here’s a grapher that shows the behavior of a rational function made up of a second degree polynomial divided by a first degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b Root of denominator: d 2nd Degree / 2nd Degree Here’s a grapher that shows the behavior of a rational function made up of a second degree polynomial divided by a second degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b Roots of denominator: d, e 2nd Degree / 3rd Degree Here’s a grapher that shows the behavior of a rational function made up of a second degree polynomial divided by a third degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b Roots of denominator: d, e, f 3rd Degree / 1st Degree Here’s a grapher that shows the behavior of a rational function made up of a third degree polynomial divided by a first degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b, c Root of denominator: d 3rd Degree / 2nd Degree Here’s a grapher that shows the behavior of a rational function made up of a third degree polynomial divided by a second degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b, c Roots of denominator: d, e

Exponential Functions Exponential functions are functions with x as the input variable, and x is in the position of an exponent to a base. That is, these functions, in simple form, look like this: f(x) = 3x The following links explain several aspects of exponential functions.

Definition Transformations of f(x) = 2x Exponential Growth Radioactive Decay Money Matters

Simple Interest Compound Interest Continuous Interest Effective Annual Rate Ordinary Annuity Loans

Transform of f(x) = 2x This Java applet demonstrates several transforms of the function f(x) = 2x. You can assign different values to a, b, h, and k and watch how these changes affect the shape of the graph.

Set a = 1.0, b = 1.0, h = 0.0 and k = 0.0 to see the basic function f(x) = 2x.

Simple Interest Explanation Simple interest is earned on the principal only. Simple interest is not an example of an exponential function. Understanding simple interest, though, will shed light on the understanding of compound interest which is an example of an exponential function.

Formula T Final value of investment P Initial value of investment n

Number of interest periods, usually number of years

r

Percentage rate per interest period, if per year, the annual percentage rate or APR

At start:

T=P After 1 period: T = P + Pr = P(1 + 1r) After 2 periods: T = P + Pr + Pr = P + 2Pr = P(1 + 2r) After 3 periods: T = P + Pr + Pr + Pr = P + 3Pr = P(1 + 3r) After n periods: T = P(1 + nr)

Example calculation If $200 dollars is invested and earns 8.0% simple interest, what is the final value of the investment after 6 years? T = P(1 + nr)

T = 200(1 + (6)(0.080)) T = $296 Calculator for simple interest T = P(1 + nr)

T Final value of investment P Initial value of investment n

Number of interest periods, usually number of years

r

Percentage rate per interest period, if per year, the annual percentage rate (APR)

Compound Interest

Explanation Compound interest is paid both on the original principal and on the accumulated past interest.

Formula S Final value of investment P Initial value of investment i Interest rate per period n Number of interest periods At start:

S=P After 1 period: S = P + Pr = P(1 + i) Amount at the start of this period Amount of interest earned during this period on the starting amount of this period Final value at end of this period After 2 periods:

S = P(1 + i) + P(1 + i)i = P(1 + i)(1 + i) = P(1 + i)2 Amount at the start of this period, i.e., the final value at the end of the last period

Amount of interest earned during this period on the starting amount of this period Final value at end of this period After 3 periods:

S = P(1 + i)2 + P(1 + i)2i = P(1 + i)2(1 + i) = P(1 + i)3 Amount at the start of this period, i.e., the final value at the end of the last period Amount of interest earned during this period on the starting amount of this period Final value at end of this period After n periods:

S = P(1 + i)n Often the interest per period, i, is expressed in terms of the annual percentage rate (APR), r, and the number of interest periods per year, k. Under these conditions the interest per period is equal to the annual percentage rate divided by the number of interest periods per year, as in: i=r/k This would make the above formula for the final value of an investment after n interest periods look like this:

S = P(1 + r/k)n Notice that the output, S, is an exponential function of n. That is, if we consider the final value of the investment as a function of the length of time for the investment, then n, the length of time for the investment, is in the exponent position, and this makes S an exponential function of n.

Example calculation If $4000 is invested at an annual rate of 6.0% compounded monthly, what will be the final value of the investment after 10 years? Since the interest is compounded monthly, there are 12 periods per year, so, k = 12. Since the investment is for 10 years, or 120 months, there are 120 investment periods, so, n = 120. S = P(1 + r/k)n S = 4000(1 + 0.06/12)120 S = 4000(1.005)120 S = 4000(1.819396734)

S = $7277.59 Calculator for compound interest

S = P(1 + r/k)n

S

Final value of investment

P

Initial value of investment

r

Annual percentage rate (APR)

k

Interest periods per year

r/k Percentage rate per interest period n

Number of interest periods

Continuous Interest

Explanation Continuous interest is a form of compound interest. With continuous interest the length of the compounding period is reasoned to be infinitely small. The interest, therefore, is compounded continuously.

Formula S Final value of investment P Initial value of investment r Annual percentage rate (APR) t Number of years Value of investment after t years:

S = Pert Where e is the transcendental number 2.7182818285... Notice that the output, S, is an exponential function of t. That is, if we consider the final value of the investment as a function of the length of time for the investment, then t, the length of time for the investment, is in the exponent position, and this makes S an exponential function of t.

Example calculation If $4000 is invested at an annual rate of 6.0% compounded continuously, what will be the final value of the investment after 10 years? S = Pert

S = 4000e(0.06)(10) S = 4000e0.6 S = 4000(1.822188)

S = $7288.48 Calculator for continuous interest:

S = Pert S Final value of investment P Initial value of investment r APR, Annual percentage rate t Number of years Effective Annual Rate

Explanation The effective annual rate is a value used to compare different interest plans. If two plans were being compared, the interest plan with the higher effective annual rate would be considered the better plan. The interest plan with the higher effective annual rate would be the better earning plan. For every compounding interest plan there is an effective annual rate. This effective annual rate is an imagined rate of simple interest that would yield the same final value as the compounding plan over one year.

Formula S

Final value of investment

P

Initial value of investment

ieff Effective annual rate r

Annual percentage rate (APR)

k

Interest periods per year

After a term of one year the final value, S, of a compounded interest investment with an initial value P compounded k times per year at an annual percentage rate of r is given by:

S = P(1 + r/k)k After a term of one year the final value, S, of a simple interest investment with an initial value P at an annual percentage rate of ieff is given by: S = P(1 + ieff) Setting these two equal we get:

P(1 + ieff) = P(1 + r/k)k Dividing each side by P we get:

(1 + ieff) = (1 + r/k)k And solving for ieff we get: ieff = (1 + r/k)k - 1

Example calculation Which plan is the better investment plan? Plan 1: 8.0% annual percentage rate, compounded monthly Plan 2:

7.9% annual percentage rate, compounded daily ieff for plan 1: ieff = (1 + r/k)k - 1 ieff = (1 + 0.08/12)12 - 1 ieff = (1.006666666)12 - 1 ieff = (1.082999498) - 1 ieff = 0.082999498 ieff = 8.2999% ieff for plan 2: ieff = (1 + r/k)k - 1 ieff = (1 + 0.079/365)365 - 1 ieff = (1.000216438)365 - 1 ieff = (1.082194930) - 1 ieff = 0.082194930 ieff = 8.2194%

Plan 1 is better since 8.2999% > 8.2194%

Calculator for effective annual rate: ieff = (1 + r/k)k - 1

ieff Effective annual rate r

APR, Annual percentage rate

k

Times per year compounded

Ordinary Annuity

Calculator for the future value of an ordinary annuity:

S = R(((1 + i)n - 1) / i) S Final value of annuity R Amount of payment i

Interest rate per period

n Number of payments

Trigonometry Realms sin(x)

Overview •

Quick view of definitions and graphs of trigonometric functions.

Introduction • •

Detailed explanation relating the basic trigonometric functions to the right triangle. Multiple choice problems concerning introductory trigonometry.

Common Angles in a Circle •

Values in degrees and radians for several usual angles from 0 to 360 degrees, (or 0 to 2Pi radians).

Point Definitions for Trig Functions •

Definitions of trig functions based on an (x, y) point on the terminal side of the input angle to the function.

Degrees, Minutes, and Seconds •

How to go back and forth between a decimal number representation of an angle’s measurement and a representation of the measurement in degrees, minutes, and seconds.

The Radian •

Often in mathematics angles are measured in radians rather than in degrees. This section explains the radian and its relationship to the degree.

Solvers •

Right Triangle Solvers

SineThe Graph of the Sine Function

The Right Triangle and the Sine Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The sine of angle A equals the length of the opposite side divided by the length of the hypotenuse. We could write:

sin(A) = opp / hyp So, for example, if the length of the opposite side was 6 and the length of the hypotenuse was 10, then we would write: sin(A) = 6 / 10 sin(A) = 0.6000

CosineThe Graph of the Cosine Function

Above is the graph of the cosine function.

The Right Triangle and the Cosine Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The cosine of angle A equals the length of the adjacent side divided by the length of the hypotenuse. We could write:

cos(A) = adj / hyp So, for example, if the length of the adjacent side was 8 and the length of the hypotenuse was 10, then we would write: cos(A) = 8 / 10 cos(A) = 0.8000

TangentThe Graph of the Tangent Function

The Right Triangle and the Tangent Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The tangent of angle A equals the length of the opposite side divided by the length of the adjacent side. We could write:

tan(A) = opp / adj So, for example, if the length of the opposite side was 6 and the length of the adjacent side was 8, then we would write: tan(A) = 6 / 8 tan(A) = 0.7500

CosecantThe Graph of the Cosecant Function

The Right Triangle and the Cosecant Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The cosecant of angle A equals the length of the hypotenuse divided by the length of the opposite side. We could write:

csc(A) = hyp / opp So, for example, if the length of the hypotenuse was 10 and the length of the opposite side was 6, then we would write: csc(A) = 10 / 6 csc(A) = 1.6667

SecantThe Graph of the Secant Function

The Right Triangle and the Secant Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The secant of angle A equals the length of the hypotenuse divided by the length of the adjacent side. We could write:

sec(A) = hyp / adj So, for example, if the length of the hypotenuse was 10 and the length of the adjacent side was 8, then we would write: sec(A) = 10 / 8 sec(A) = 1.2500

CotangentThe Graph of the Cotangent Function

The Right Triangle and the Cotangent Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The cotangent of angle A equals the length of the adjacent side divided by the length of the opposite side. We could write:

cot(A) = adj / opp So, for example, if the length of the adjacent side was 8 and the length of the opposite side was 6, then we would write: cot(A) = 8 / 6 cot(A) = 1.3333 Trigonometry and Right Triangles First of all, think of a trigonometry function as you would any general function. That is, a value goes in and a value comes out. If that does not seem quite clear, go see The Definition of a Function and What is f(x)? The names of the three primary trigonometry functions are:

sine cosine tangent These are abbreviated this way:

sine.....sin cosine.....cos tangent.....tan So, instead of writing f(x) , we will write:

sin(x) cos(x) tan(x) Often, in general mathematics notation the parentheses are dropped from the above examples. Therefore, the notation will often look like this:

sin x cos x tan x In Zona Land we will keep the parentheses.

Now, as usual, the input value is x. This input value usually represents an angle. For the sine function, when the input value is 30 degrees, the output value is 0.5. We would write that statement this way: 0.5 = sin(30 °) Below is a listing of several popular input and output values for the three main trigonometry functions. You do not have work at memorizing this table. After you use trigonometry for a while, these values will be remembered quite easily.

0.0000 = sin(0 °)

1.0000 = cos(0 °)

0.0000 = tan(0 °)

0.5000 = sin(30 °) 0.8660 = cos(30 °) 0.5773 = tan(30 °) 0.7071 = sin(45 °) 0.7071 = cos(45 °) 1.000 = tan(45 °) 0.8660 = sin(60 °) 0.5000 = cos(60 °) 1.7320 = tan(60 °) 1.0000 = sin(90 °) 0.0000 = cos(90 °) +infinity = tan(90 °) At this point our central issues will revolve around these questions:

Where do these numbers come from?

What do these numbers mean? Why, for example, is the cosine of 30 degrees equal to 0.8660? The input value for these trigonometric functions is an angle. That angle could be measured in degrees or radians. Here we will consider only input angles measured in degrees from 0 degrees to 90 degrees. This input value appears within the parentheses throughout the above table. The output value for these trigonometric functions is a pure number. That is, it has no unit. This output value appears to the left of the equal sign throughout the above table. There are several ways to understand why a certain input angle produces a certain output value. At first, the most important manner of understanding this is tied to right triangles. All of the trigonometric values for angles between 0 degrees and 90 degrees can be understood by considering this diagram:

We will be concerned angle A. Notice that the sides of the triangle are labeled appropriately “opposite side” and “adjacent side” relative to angle A. The hypotenuse is not considered opposite or adjacent to the angle A. We will also be concerned with length of the three sides. For this discussion we will call the “length of the opposite side” simply the “opposite”. Similarly, the other two lengths will be called “adjacent” and “hypotenuse”. The value for the sine of angle A is defined as the value that you get when you divide the opposite side by the hypotenuse. This can be written: sin(A) = opposite / hypotenuse Or simply: sin(A) = opp / hyp Or, even more simply: sin(A) = o / h Suppose we measure the lengths of the sides of this triangle. Here are some realistic values:

This would mean that: sin(A) = opposite / hypotenuse = 4.00 cm / 7.21 cm = 0.5548 Or simply: sin(A) = 0.5548 Now for the other two trig functions. The value for the cosine of angle A is defined as the value that you get when you divide the adjacent side by the hypotenuse. This can be written: cos(A) = adjacent / hypotenuse Or: cos(A) = adj / hyp Or: cos(A) = a / h Using the above measured triangle, this would mean that: cos(A) = adjacent / hypotenuse = 6.00 cm / 7.21 cm = 0.8322 Or simply: cos(A) = 0.8322 The value for the tangent of angle A is defined as the value that you get when you divide the opposite side by the adjacent side. This can be written: tan(A) = opposite / adjacent Or: tan(A) = opp / adj

Or: tan(A) = o / a Using the above measured triangle, this would mean that: tan(A) = opposite / adjacent = 4.00 cm / 6.00 cm = 0.6667 Or simply: tan(A) = 0.6667 The angle A in the above triangle is actually very close to 33.7 degrees. So, we would say:

0.5548 = sin(33.7°) 0.8322 = cos(33.7°) 0.6667 = tan(33.7°) So, suppose that you wanted to know the trigonometry values for 47.5 degrees? You could carefully draw a right triangle using a ruler and protractor that had an angle equal to 47.5 degrees in the position of angle A. Then, you could carefully measure the sides. Lastly you could divide the appropriate sides to find the values for the three trigonometric functions. You would find that:

0.7373 = sin(47.5°) 0.6755 = cos(47.5°) 1.0913 = tan(47.5°) Someone has already done this, in a way, for all the possible angles. All the input angles and output values are listed in tables called trig tables. They look like this:

Angle

sin

cos

tan

0.0

0.0000

1.0000 0.0000

0.5

0.0087

0.9999 0.0087

1.0

0.0174

0.9998 0.0174

And so on...

These let you look up the trigonometric values for any angle. Calculators and computers, of course, will let you do the same.

Here is a demonstration that shows you these trig calculations for several angles. Use the slider to adjust the size of the angle. Notice how the values are calculated for each trig function depending upon the lengths of the sides of the triangle. Below is again the triangle from the above diagrams, except now the other acute angle, B, is marked. Also marked are the sides that are adjacent and opposite to angle B.

Here are the three trig functions for angle B:

sin(B) = opposite / hypotenuse = 6.00 cm / 7.21 cm = 0.8322 cos(B) = adjacent / hypotenuse = 4.00 cm / 7.21 cm = 0.5548 tan(B) = opposite / adjacent = 6.00 cm / 4.00 cm = 1.5000 If you look carefully you will notice that the sine of angle B is the same value we calculated above for the cosine of angle A. You should also notice that the cosine of angle B is equal to previous calculation for the sine of angle A. This is because the opposite side for angle B is the adjacent side for angle A, and because the adjacent side for angle B is the opposite side for angle A. This is demonstrated in the following diagram:

We could summarize this relationship this way:

sin(A) = A’s opposite / hypotenuse = 4.00 cm / 7.21 cm = 0.5548 cos(B) = B’s adjacent / hypotenuse = 4.00 cm / 7.21 cm = 0.5548 cos(A) = A’s adjacent / hypotenuse = 6.00 cm / 7.21 cm = 0.8322 sin(B) = B’s opposite / hypotenuse = 6.00 cm / 7.21 cm = 0.8322 Now, angle A and B form a pair of complementary angles. That is, their measurements add up to 90 degrees. This is because the measurement of the interior angles for any triangle must sum to 180 degrees, and in this triangle 90 of those degrees are taken up by the right angle, so that leaves 90 degrees remaining from the total of 180 to be split up between angle A and B. So, here we notice that the sine of an angle is equal to the cosine of its complement, and that the cosine of an angle is equal to the sine of its complement. Also, we will take note of the relationship between the tangents of the complementary angles A and B. The tangent of angle A is equal to the reciprocal, or inverse, of the tangent of angle B, and, likewise, the tangent of angle B is equal to the reciprocal of the tangent of its complement, angle A. This is summarized in the following table:

tan(A) = A’s opposite / A’s adjacent = 4.00 cm / 6.00 cm = 0.6667 tan(B) = B’s opposite / B’s adjacent = 6.00 cm / 4.00 cm = 1.5000 Here is an easy way to remember these relationships for trig functions and the right triangle. Just write down this mnemonic: SOH - CAH - TOA It is pronounced “so - ka - toe - ah”. The SOH stands for “Sine of an angle is Opposite over Hypotenuse.”

The CAH stands for “Cosine of an angle is Adjacent over Hypotenuse.” The TOA stands for “Tangent of an angle is Opposite over Adjacent.” What is f(x) ?

Common Angles Around a Circle The circle below can be thought of as being divided into angles that are integer multiples of 30, 45, 60, and 90 degree angles. Click one of the points on the circle to see the angle and its measurement in both degrees and radians. See notes below. The angles marked on this circle represent common angles that are often used in introductory geometry and trigonometry problems. Try figuring out the angle measurement before you click on the point. Then use this applet to check your value. All of the angles will be shown as arcs when they are drawn. You should notice that all of these angles are in standard position. This Java applet will not print the value for pi as a Greek letter. Instead, it is printed as ‘Pi’. This, of course, approximately equals the value 3.14. Trig Function Point Definitions Quick Instructions: • •

The above applet starts with the sine function active. Any of the six trigonometric functions can be activated by choosing the appropriate radio button at the top of the applet. The large square graph on the left is the (x, y) coordinate plane. It extends from 10.0 to +10.0 along the x-axis and the y-axis.

•

• •

•

•

The upper right rectangular graph shows the graph of the currently active trig function. For example, when the sine function is active, this graph shows the sin(a) vs. a, where a represents the angle. Horizontally, this graph has a domain from 0 radians to 2pi radians with markers every pi/2 radians. Vertically, the range of this graph runs from -2.0 to +2.0 for the sine and cosine functions, and it runs from -5.0 to +5.0 for the tangent, cotangent, secant and cosecant functions. Vertical marking occur every unit distance. The lower right rectangular area shows the calculation for the currently active trig function at the current (x, y) point on the left graph. When the sine function radio button is selected, click somewhere on the left (x, y) graph. Notice: o The (x, y) coordinates are presented above the selected point. o The angle whose terminal side goes through this (x, y) point is drawn. Its value is presented in the upper right corner of this (x, y) graph. o The relevant quantities necessary for the sine calculation are drawn on the graph, (the y distance and the radius in this case). The values for these quantities are presented near their graphic representations. o The upper right hand graph shows the current function, i. e., the sine function. The current input angle and the current output value for this function are presented. o In the lower right rectangle of the applet the value for the current trig function at the current angle is calculated using the the relevant quantities from the (x, y) graph. All values are rounded to two decimal places. Certainly more precise values for the trig functions are available elsewhere. This applet, though, is not meant to be a calculator. It is meant to demonstrate the interrelationships of several trigonometric concepts. Try different (x, y) positions and different trig functions.

Further Discussion: This material explains the definitions of the six trigonometric functions in terms of an (x, y) point located on the terminal side of the input angle. You should be familiar with: • The (x, y) coordinate plane. • How to find the distance from the origin to an (x, y) point. • The graphs of the trig functions. • Angles in standard position. • Radian measure for angles. The six trigonometric functions, (sine, cosine, tangent, cotangent, secant and cosecant), are usually thought to accept an angle as input and output a pure number. For the purposes of the definitions this angle is to be placed in standard position. We will be concerned with any (x, y) point located on the terminal side of this angle. These definitions are based on such an (x, y) point. These definitions also use the distance from the origin to the (x, y) point. This distance will be referred to as r and can be calculated like this:

The six definitions are: sin(angle) = y/r cos(angle) = x/r tan(angle) = y/x (x not equal to zero) csc(angle) = r/y (y not equal to zero) sec(angle) = r/x (x not equal to zero) cot(angle) = x/y (y not equal to zero) So, for example, the point (5, 7) is on the terminal side of an angle in standard position which has a measure of about 0.95 radians (about 54 degrees):

The distance from the origin to the point (5, 7) can be calculated this way (approximate result given):

Therefore, the six trigonometric function values for this angle can be calculated as follows (approximate results given):

sin(0.95) = y/r = 7/8.6 = 0.81 cos(0.95) = x/r = 5/8.6 = 0.58 tan(0.95) = y/x = 7/5 = 1.4

csc(0.95) = r/y = 8.6/7 = 1.2 sec(0.95) = r/x = 8.6/5 = 1.7 cot(0.95) = x/y = 5/7 = 0.71 In all of the above calculations approximate results were given. Of course, if you are doing more careful and important work you would use measurements and calculations of higher significance. The definitions given here for the six trigonometric functions are more powerful than the right triangle definitions given in the introduction to trigonometry section. The right triangle definitions are only good for angles up to pi/2 radians (90 degrees). These trig definitions based upon an (x, y) point on the terminal side of an angle are good for angles of any measurement, positive or negative.

Degrees, Minutes, Seconds

There are several ways to measure the size of an angle. One way is to use units of degrees. (Radian measure is another way.)

In a complete circle there are three hundred and sixty degrees. An angle could have a measurement of 35.75 degrees. That is, the size of the angle in this case would be thirty-five full degrees plus seventy-five hundredths, or three fourths, of an additional degree. Notice that here we are expressing the measurement as a decimal number. Using decimal numbers like this one can express angles to any precision - to hundredths of a degree, to thousandths of a degree, and so on. There is another way to state the size of an angle, one that subdivides a degree using a system different than the decimal number example given above. The degree is divided into sixty parts called minutes. These minutes are further divided into sixty parts called seconds. The words minute and second used in this context have no immediate connection to how those words are usually used as amounts of time.

In a full circle there are 360 degrees. Each degree is split up into 60 parts, each part being 1/60 of a degree. These parts are called minutes. Each minute is split up into 60 parts, each part being 1/60 of a minute. These parts are called seconds.

The size of an angle could be stated this way: 40 degrees, 20 minutes, 50 seconds. There are symbols that are used when stating angles using degrees, minutes, and seconds. Those symbols are show in the following table.

Symbol for degree: Symbol for minute: Symbol for second: So, the angle of 40 degrees, 20 minutes, 50 seconds is usually written this way:

How could you state the above as an angle using common decimal notation? The angle would be this many degrees, (* means times.):

40 + (20 * 1/60) + (50 * 1/60 * 1/60) That is, we have 40 full degrees, 20 minutes - each 1/60 of a degree, and 50 seconds each 1/60 of 1/60 of a degree. Work that out and you will get a decimal number of degrees. It’s 40.34722... Going the other way is a bit more difficult. Suppose we start with 40.3472 degrees. Can we express that in units of degrees, minutes, and seconds? Well, first of all there are definitely 40 degrees full degrees. That leaves 0.3472 degrees. So, how many minutes is 0.3472 degrees? Well, how many times can 1/60 go into 0.3472? Here’s the same question: What is 60 times 0.3472? It’s 20.832. So, there are 20 complete minutes with 0.832 of a minute remaining. How many seconds are in the last 0.832 minutes. Well, how many times can 1/60 go into 0.832, or what is 60 times 0.832? It’s 49.92, or almost 50 seconds. So, we’ve figured that 40.3472 degrees is almost exactly equal to 40 degrees, 20 minutes, 50 seconds. (The only reason we fell a bit short of 50 seconds is that we really used a slightly smaller angle in this second half of the calculation explanation. In the original angle, 40.34722... degrees, the decimal repeats the last digit of 2 infinitely, so, the original angle is a bit bigger than 40.3472.)

The Radian The word radian describes a certain size of an angle. In the Java animation below the blue pie slice shape demonstrates an angle of one radian. Notice that for an angle of one radian the arc length along the edge of the circle is equal in length to the radius. Read the details below... Usually, a person first learns how to measure the size of an angle using degrees. There are, of course, 360 degrees all the way around a circle. The degree, however, turns out to be not a very mathematical way of measuring angles. Perhaps the degree was originally created as approximately the angle which the Earth goes through per day as it orbits the Sun, since there are about 360 days in the year. The radian is used much more than the degree in higher mathematics for measuring angles. A radian is defined this way: • •

If you have a circle with an angle whose vertex is at the center of that circle... And if that angle is of such a size that the amount of the circumference of the circle which that angle intercepts has an arc length equal to the length of the radius...

• Then that angle has a measurement of one radian. Here’s another way to say it: A central angle to a circle has a size of one radian if it subtends an arc length on the circle equal in length to the radius. In other words, if you could pick up the radius of a certain circle like it were a plastic rod and bend it around the circumference of that circle, then that bent radius length would touch the sides of a central angle which had a measurement of one radian. Look at the following picture and then go back and look at the animation.

One thing to understand about a radian is that it is bigger than a degree. In fact: 1 radian = 57.2957 degrees 1 degree = 0.0174532 radians The above values are to six significant figures, truncated, not rounded. If you want to know the exact size of the radian in terms of degrees, take 360 and divide it by 2 times pi. That number is how many degrees there are in a radian. What is the reasoning behind these values? It works this way: How many times can you trace an arc length equal to the size of the radius as you move around the circumference of a circle? Well, here’s the formula for the circumference of a circle:

C is the circumference, and r is the radius. Looks like there are two pi radii around the circumference of a circle, or about 2 times 3.14, i.e., about 6.28, radii around the circumference of a circle. Each one of these radius lengths would designate one radian, so there are about 6.28 radians in a full circle. The following diagram shows this:

Therefore, there are 2 pi radians in a full circle. We also know that there are 360 degrees in a circle. So, there are 360 degrees per 2 pi radians. Dividing 360 by 2 pi give us the value of about 57.2957 degrees per radian. A radian is equal to 57.2957 degrees. Also, by dividing 2 pi radians by 360 degrees we get about 0.0174532 radians per degree. A degree is equal to 0.0174532 radians. In mathematics if you state the size of an angle as a pure number, without the degree ‘unit’ marker after it, then the angle is taken to be in radians. So, if the angle in question is named A, and if someone were to write down:

A=4 Then that would mean that angle A has a measurement of 4 radians and not 4 degrees. Trigonometric Functions - Right Triangle Solvers Here is a group of solvers that work with the following right triangles.

In each of the above diagrams the acute angles of the right triangle are named angle D and angle E. The right angle is not named. The two sides, or legs, of the triangle have lengths named d and e. The length of the hypotenuse is named f. In each case you will be starting with known values for two parts of the right triangle. For example, in the first triangle you are starting with known values for angle D and side length d. You are to create these initial known values. In each case you are to find the other parts of the right triangle. For example, in the first triangle you are to find values for angle E and the lengths of sides d and e and the length of the hypotenuse f.

Intersections

Intersection of Two Lines Intersection of a Line and a Parabola Intersection of Two Parabolas

Intersection of Two Lines

Here we will cover a method for finding the point of intersection for two linear functions. That is, we will find the (x, y) coordinate pair for the point were two lines cross. Our example will use these two functions: f(x) = 2x + 3 g(x) = -0.5x + 7 We will call the first one Line 1, and the second Line 2. Since we will be graphing these functions on the x, y coordinate axes, we can express the lines this way: y = 2x + 3 y = -0.5x + 7 Those two lines look this way:

Now, where the two lines cross is called their point of intersection. Certainly this point has (x, y) coordinates. It is the same point for Line 1 and for Line 2. So, at the point of intersection the (x, y) coordinates for Line 1 equal the (x, y) coordinates for Line 2.

Since at the point of intersection the two y-coordinates are equal, (we’ll get to the xcoordinates in a moment), let’s set the y-coordinate from Line 1 equal to the y-coordinate from Line 2. The y-coordinate for Line 1 is calculated this way: y = 2x + 3 The y-coordinate for Line 2 is calculated this way: y = -0.5x + 7 Setting the two y-coordinates equal looks like this:

2x + 3 = -0.5x + 7 Now, we do some algebra to find the x-coordinate at the point of intersection:

2x + 3 = -0.5x + 7

We start here.

2.5x + 3 = 7

Add 0.5x to each side.

2.5x = 4

Subtract 3 from each side.

x = 4/2.5

Divide each side by 2.5.

x = 1.6

Divide 4 by 2.5.

So, we have the x-coordinate for the point of intersection. It’s x = 1.6. Now, let’s find the y-coordinate. The y-coordinate can be found by placing the x-coordinate, 1.6, into either of the equations for the lines and solving for y. We will first use the equation for Line 1:

y = 2x + 3 y =2(1.6) + 3 y = 3.2 + 3 y = 6.2 Therefore, the y-coordinate is 6.2. To make sure our calculations are correct, and also to demonstrate a point, we should get the same y-coordinate if we use the equation for Line 2. Let’s try that: y = -0.5x + 7 y = -0.5(1.6) + 7 y = -0.8 + 7 y = 6.2 Well, looks like everything has worked out. The point of intersection for these two lines is (1.6, 6.2). If you look back at the graph, this certainly makes sense:

Here’s the summary of our methods: 1. Get the two equations for the lines into slope-intercept form. That is, have them in this form: y = mx + b. 2. Set the two equations for y equal to each other. 3. Solve for x. This will be the x-coordinate for the point of intersection. 4. Use this x-coordinate and plug it into either of the original equations for the lines and solve for y. This will be the y-coordinate of the point of intersection. 5. As a check for your work plug the x-coordinate into the other equation and you should get the same y-coordinate. 6. You now have the x-coordinate and y-coordinate for the point of intersection. Actually, there is nothing special about the functions being linear functions. This method could be used to find the point or points of intersection between many other types of functions. One would express the functions in ‘y =’ form, set the right side of these forms equal to each other, solve for x, (or x’s), and use this x, (or x’s), to find the corresponding y, (or y’s).

Intersection of Line and Parabola

Here we will cover a method for finding the point or points of intersection for a linear function and a quadratic function. Quadratic functions graph as parabolas. So, we will find the (x, y) coordinate pairs where a line crosses a parabola. First, understand that a line and a parabola may intersect at two points, as in this picture:

Or, they may intersect at only one point, as in this picture:

And, they may not meet at any points, as shown here:

For this example we will use these two functions: f(x) = 1.5x + 5 g(x) = 2x2 + 12x + 13 The first one, f(x), is a line, and the second, g(x), is a parabola. Since we will be graphing these functions on the x, y coordinate axes, we can express them this way: y = 1.5x + 5 y = 2x2 + 12x + 13 When graphed, these two functions look like this:

Now, the places where the two functions cross are called their points of intersection. At these points of intersection the x-coordinate for the line equals the x-coordinate for the parabola, and the y-coordinate for the line equals the y-coordinate for the parabola.

Since at the points of intersection the y-coordinates are equal, (we’ll get to the xcoordinates in a moment), let’s set the y-coordinate from line equal to the y-coordinate from parabola. The y-coordinate for the line is calculated this way: y = 1.5x + 5 The y-coordinate for the parabola is calculated this way: y = 2x2 + 12x + 13 Setting the two y-coordinates equal looks like this:

1.5x + 5 = 2x2 + 12x + 13 When we solve the above equation, we find the x-coordinates for the points of intersection. Here’s the algebra:

We start here. Subtract 1.5x from each side. Subtract 5 from each side. Use the quadratic formula to solve for x. Here are are two solutions for x. Now, we will plug each value for the x-coordinate into either of the intersecting functions to get its corresponding y-coordinate. Let’s first work with x = -0.92. We will plug this into the equation of the line to get the y-coordinate. Here is that work: y = 1.5x + 5 y = 1.5(-0.92) + 5 y = -1.38 + 5 y = 3.62 Notice that if we would have chosen to have worked with the quadratic function, we would have gotten the same y-coordinate. (We are off just a bit due to rounding and truncating decimals along the way, both above in the evaluation of the quadratic formula and below where we keep intermediate results to two decimals.): y = 2x2 + 12x + 13 y = 2(-0.92)2 + 12(-0.92) + 13 y = 1.69 - 11.04 + 13 y = 3.65 So, to a fair degree of accuracy, the two functions intersect at (-0.92, 3.62). Let’s find the other point of intersection. We know that the x-coordinate is -4.33, and using the equation for the line we can find the y-coordinate: y = 1.5x + 5 y = 1.5(-4.33) + 5 y = -6.49 + 5 y = -1.49

Therefore, the two functions also intersect at (-4.33, -1.49). Here’s how those spots look on the graph:

You can use the above procedure can be used to find the intersection of any line with any parabola. Of course, the line and parabola will not always intersect at two points. Sometimes they will only intersect at one point, and quite often they will not intersect at all. These conditions will show up when you solve the quadratic equation after you set the two separate functions equal to each other and collect like terms. When solving that quadratic equation, if the discriminant equals zero, then you will have only one solution for x, which boils down to only one point of intersection. If the discriminant is negative, then there are no solutions for x, which means the two parabolas do not intersect. Want to try this yourself? Below is a calculator which you can use to check your work. First, make up two functions, one for a line and one for a parabola. You could use the function graphers in the Function Institute to help you find the values for the first degree polynomial, (linear function for the line), and the second degree polynomial, (quadratic function for the parabola), or you could just make them up. Perhaps you would choose: y = 1.5x + 5 (for the line) y = 2x2 - 12x + 13 (for the parabola) If you wanted to see these functions using EZ Graph you would type each into one of the ‘y = ‘ boxes at the bottom of the grapher like this: 1.5*x + 5 2*x^2 - 12*x + 13 Intersection of Two Parabolas

We will cover a method for finding the point or points of intersection for two quadratic functions. Quadratic functions graph as parabolas. So, we will find the (x, y) coordinate pairs where the two parabolas intersect. First, understand that two parabolas may intersect at two points, as in these pictures:

Or, the parabolas may intersect at only one point, as in these pictures:

And, they may not meet at any points, as shown here:

Our example will use these two quadratic functions: f(x) = 1.5x2 - 9x + 11.5 g(x) = -0.2x2 - 0.4x + 2.8 Of course, quadratic functions, or second degree polynomial functions, graph as parabolas. Since we will be graphing these functions on the x, y coordinate axes, we can express the parabolas this way: y = 1.5x2 - 9x + 11.5 y = -0.2x2 - 0.4x + 2.8 Those two parabolas look this way:

Now, where the two parabolas cross is called their points of intersection. Certainly these points have (x, y) coordinates, and at the points of intersection both parabolas share the same (x, y) coordinates. So, at the points of intersection the (x, y) coordinates for f(x) equal the (x, y) coordinates for g(x).

Since at the points of intersection the y-coordinates are equal, (we’ll be working with the x-coordinates later), let’s set the y-coordinate from from one parabola equal to the ycoordinate from the other parabola. The y-coordinate for f(x) is calculated this way: y = 1.5x2 - 9x + 11.5 The y-coordinate for g(x) is calculated this way: y = -0.2x2 - 0.4x + 2.8 Setting the two y-coordinates equal to each other creates this equation:

1.5x2 - 9x + 11.5 = -0.2x2 - 0.4x + 2.8 Now, we do some algebra to find the x-coordinates at the points of intersection:

This is where we start. Subtract 1.5x2 from each side. Add 9x to each side. Subtract 11.5 from each side. This leaves us with a quadratic equation. Use the quadratic formula to solve for x.

These are the xcoordinates for the two points of intersection. (The values her are truncated to two decimal places and not rounded.) So, we have the x-coordinates for the points of intersection. They are x = 1.39 and x = 3.66. Now, let’s find the y-coordinates. Each y-coordinate can be found by placing its corresponding x-coordinate into either of the equations for the parabolas and solving for y. We will first use the equation from the first parabola. Here we plug x = 1.39 into it: y = 1.5x2 - 9x + 11.5 y = 1.5(1.39)2 - 9(1.39) + 11.5 y = 2.89 - 12.51 + 11.5 y = 1.88 So, one point of intersection is very close to (1.39, 1.88). Here, we say very close to since the values have been calculated using only two decimal places. Now, plugging in the other x-coordinate into the equation for the first parabola, we can get the other y-coordinate for the second point of intersection: y = 1.5x2 - 9x + 11.5 y = 1.5(3.66)2 - 9(3.66) + 11.5 y = 20.09 - 32.94 + 11.5 y = -1.35 The other point of intersection is very near (3.66, -1.35). Here are these points of intersection shown on the graph of the two parabolas:

The above procedure can be used to find the intersection of any two parabolas. Of course, the parabolas will not always intersect at two points. Sometimes they will only intersect

at one point, and quite often they will not intersect at all. These conditions will show up when you solve the quadratic equation after you set the two separate functions equal to each other and collect like terms. When solving that quadratic equation, if the discriminant equals zero, then you will have only one solution for x, which boils down to only one point of intersection. If the discriminant is negative, then there are no solutions for x, which means the two parabolas do not intersect. Ready to try it yourself? Below is a calculator which you can use to check your work. First, make up two second degree polynomial functions, (quadratic functions), in this form: y = ax2 + bx + c You could use the function grapher in the Function Institute to help you find the values for a, b, and c, or you could just make them up. Perhaps for one of your parabolas you would choose: y = 1.5x2 - 9x + 11.5 If you wanted to see this function using EZ Graph you would type this into one of the ‘y = ‘ boxes at the bottom: 1.5*x^2 - 9*x + 11.5

The Geometry Section

Common Shapes Areas and Volumes Coordinate Geometry Points, Lines, Planes

Common Shapes

Two Dimensional Circle Rhombus Trapezoid The Circle

The circle is a set of points equally distant from one central point. This central point is called the center of the circle. Here the center is shown with a small cross. The distance from the center to the curve of the circle is called the radius. Here the radius is shown as a line from the center to the curve of the circle.

The distance across the circle is called the diameter. The diameter always passes through the center of the circle. The diameter is equal in length to twice the radius.

The distance all the way around the circle is called the circumference. The circumference is usually symbolized with a ‘C’ as shown here. The length of the circumference is equal to 2 times pi times the radius. Usually, as here, the radius is symbolized with an ‘r’. The Rhombus

The rhombus is a quadrilateral with all sides equal in length. Notice that it is a parallelogram, since opposite sides are parallel.

Since the rhombus is a parallelogram, you find its area using the same method that you would use for a parallelogram. The length of the base of the rhombus is the length of one of its sides, here shown with ‘b’. The height is the perpendicular distance between opposite sides, here shown with an ‘h’. The area of the rhombus is equal to its base times its height, or: area = bh

The diagonals of a rhombus are perpendicular.

A square is a rhombus since it is a quadrilateral with all sides equal.

The Trapezoid

The trapezoid is a quadrilateral with one pair of parallel sides.

The parallel sides of a trapezoid are called the bases, here symbolized by b1 and b2. The height of the trapezoid is the perpendicular distance between the bases, here symbolized by h. The area of the trapezoid is equal to the average of the bases times the height. So, you add the two bases, divide by 2, and then multiply by the height. This would be: area = ((b1 + b2) / 2)h

If the two sides which are not parallel have equal lengths, then the trapezoid is called an isosceles trapezoid. The base angles in an isosceles trapezoid are equal in measurement.

Another example of a trapezoid.

Yet another example of a trapezoid.

Total Surface Areas and Volumes of Simple Solids Box Cylinder Box with Cylindrical Hole Total Surface Area and Volume of a Box

The total surface area is made up of three pairs of sides for a total of six sides. We must find Areahw, which is the area of the side that is h by w.

Areahw = (h)(w) We must find Areahl, which is the area of the side that is h by l. Areahl = (h)(l) We must find Areawl, which is the area of the side that is w by l. Areawl = (w)(l) Each of the above areas are present on two sides of the box. So, the total surface area is sum of twice each of these areas, that is: Total Surface Area = 2(Areahw) + 2(Areahl) + 2(Areawl) The volume of the box is simply the product of h, w, and l. Volume = (h)(w)(l) Example: Let h = 3 cm, w = 5 cm, l = 8 cm Areahw = (h)(w) = (3 cm)(5 cm) = 15 cm2 Areahl = (h)(l) = (3 cm)(8 cm) = 24 cm2 Areawl = (w)(l) = (5 cm)(8 cm) = 40 cm2 Total Surface Area = 2(Areahw) + 2(Areahl) + 2(Areawl) Total Surface Area = 2(15 cm2) + 2(24 cm2) + 2(40 cm2) Total Surface Area = 30 cm2 + 48 cm2 + 80 cm2 Total Surface Area = 158 cm2 Volume = (h)(w)(l) Volume = (3 cm)(5 cm)(8 cm) Volume = 120 cm3 Total Surface Area and Volume of a Cylinder

The total surface area is made up of a circular top and bottom, which are each the same size, and the side surface, which is actually a rectangle. We must find the area of the top, or Areatop, which is the area of a circle with a radius of r. Areatop = (pi)(r2) This area is the same as the area of the bottom, so: Areabottom = Areatop The area of the side is actually a rectangle. Imagine the cylinder to be a simple tin can. In your mind, cut off the top and the bottom and discard them. What is left is an open ended tube. Again in your mind, use some tin snips to cut along a straight line from top to bottom and uncurl and flatten out the tube. You would be looking at a rectangle that is as tall as the cylinder and as long as the circumference of its top (or bottom). Therefore, the area of the side is: Areaside = (h)(Circumferencetop) Now, since the circumference of the top is: Circumferencetop = 2(pi)® The area of the side is: Areaside = (h)(Circumferencetop) Areaside = (h)(2(pi)®) Areaside = 2(h)(pi)® So, the total area of the cylinder is equal to the top area plus the bottom area plus the side area:

Total Surface Area = Areatop + Areabottom + Areaside The volume of the cylinder, since it is a right solid, is the product of the top area times the height.

Volume = (Areatop)(h)

Example: Let h = 5 cm, r = 2 cm Areatop = (pi)(r2) Areatop = (3.14)(2 cm)2 = (3.14)4 cm2 = 12.56 cm2

Areabottom = Areatop = 12.56 cm2 Areaside = (h)(Circumferencetop) Areaside = (h)(2(pi)®) Areaside = 2(h)(pi)® Areaside = 2(5 cm)(3.14)(2 cm) Areaside = 62.8 cm2 Total Surface Area = Areatop = + Areabottom + Areaside Total Surface Area = 12.56 cm2 + 12.56 cm2 + 62.8 cm2 Total Surface Area = 87.92 cm2 Volume = (Areatop)(h) Volume = (12.56 cm2)(5 cm) Volume = 62.8 cm3 Total Surface Area and Volume of a Box with a Cylindrical Hole

Be sure you understand how to find the total surface area and volume of a box and of a cylinder before going ahead here. The total surface area of this new shape is made up several areas, some like parts of the box, some like parts of the cylinder, and some like differences between parts of the box and cylinder. Like the box, this shape has a pair of sides with areas which we will again call Areahw. Like the box, this shape also has another pair of sides with areas which we will again call Areahl. The top and bottom of this shape form another pair of equal areas. These areas are both equal to the area of the top of the box after a circular hole has been cut out of it. This hole

has an area equal to the area of the circular top of the cylinder. So, the area of the top of this shape is calculated this way:

Areatop = AreaboxTop - AreacylinderTop The area of the top of the box is Arealw, so, restating the above:

Areatop = Arealw - AreacylinderTop Again, we have two areas, top and bottom, that are equal to the immediate above value. In this shape the area of the side of the hole is like the area of the side of the cylinder, so: AreaholeSide = AreacylinderSide So, the total area of this shape is equal to: Areatotal = 2(Areahw) + 2(Areahl) + 2(Areatop) • AreaholeSide The volume of this shape is the volume of the box, as calculated before, minus the volume of the cylindrical hole, also as calculated before.

Volume = Volumebox - Volumecylinder Coordinate Geometry

VRML (x, y, z) Axes and Planes

VRML (x, y, z) Axes and Planes Worsslds These worlds illustrate the (x, y, z) coordinate axes and the three planes associated with these axes. The axes and planes all extend to infinity. However, in these demonstrations they are drawn to a finite size. Use the Back button of your browser to return from them.

Conic Sections A conic section is a section of a cone. The popular ellipse, parabola, and hyperbola, along with a few other mathematical shapes, can each be seen to be a section of a cone.

In this context, the cone is thought to be a hollow cone, quite a bit like the ‘sugar cones’ in an ice cream shop. So, these mathematical cones look like this one:

And actually when thinking about conic sections, we envision two such cones lined up vertically tip to tip. So, conic section cones in mathematics look like this:

We must think of these cones as going on forever without a top or bottom limit.

(Such a pair of cones is formally called a circular conical surface, and a discussion of conic sections is truly centered around such a surface.) A line which we imagine running through the center of the cones in a direction perpendicular to their bases is called the axis. This is shown below:

Now, about the section part in the term conic section. This section is a very thin slice of the cones. In fact, it is an infinitely thin slice. One thinks of the cones as being sliced by a plane. So, we speak about the intersection of the cones with a plane. The shape of this intersection is the shape of the conic section. Below is one example of how we could imagine a cone being intersected by a plane. (Of course, both the cone and the plane actually extend to infinity.)

The shape of the intersection, or cut, that the plane makes with the cone is the shape of the conic section. For example, in the above picture the intersection is a hyperbola. The way in which the plane cuts through the cone determines the particular conic section. It determines if the conic section is a parabola or ellipse, and so on. Several ways in which a plane can intersect the cone are illustrated in the following VRML animation. (If you have any problem viewing VRML in your browser, you probably need a VRML plug-in, something which is well worth getting. You can get help with that here.) This and the other VRML animations listed below show cones with bottoms rather than showing hollow cones. As it turns out, this makes it much easier to see cones in VRML.

Conic Sections - The Ellipse On an (x, y) graph an ellipse can have this shape:

If a plane intersects the cone all of the way around but is not perpendicular to the axis, the conic section is an ellipse. This is shown below:

Conic Sections - The Parabola On an (x, y) graph a parabola can have this shape:

If a plane intersects the cone when it is slanted the same as the side of the cone, (formally, when it is parallel to the slant height), the conic section is a parabola. This is shown below:

Conic Sections - The Hyperbola On an (x, y) graph a hyperbola can have this shape:

If a plane intersects the cone when it is parallel to the axis, the conic section is a hyperbola. This is shown below:

Conic Sections - The Circle

On an (x, y) graph a circle can have this shape:

If a plane intersects the cone when it is perpendicular to the axis, the conic section is a circle. Actually, a circle is a special case of an ellipse. This is shown below:

Conic Sections - The Line

On an (x, y) graph a line can have this shape:

If a plane intersects the cone when it is slanted the same as the side of the cone, (formally, when it is parallel to the slant height), and when it passes through the tip of the cone, (formally when it passes through the vertex), the conic section is a line. This is shown below:

Conic Sections - The Point

On an (x, y) graph a point has this shape:

If a plane intersects the cone when it is perpendicular to the axis and when it passes through the tip of the cone, the conic section is a point. Actually, in this discussion a point is a special case of a circle. This is shown below:

Simple Data Grapher Here is a very simple data grapher. Basically, you enter your legends, maximums, minimums, and data, and then you press the ‘Make graph’ button. A new browser window will open up with your graph displayed in it. Simply print the page using the services of the new browser window, and you will have a hard copy of your graph. Close the new browser window when you are done. This page remains displayed in this original browser window. Suggestions and help: • • •

Data points are plotted with this character: ‘*’. Use font size 1. Graphs made with this small font seem to print best on most equipment. The independent axis is the horizontal axis, usually called the x-axis in algebra. (Beware: In physics x usually means position and is often graphed vertically.). The dependent axis is the vertical axis, usually called the y-axis in algebra.

•

•

•

The syntax for entering data points works like this: If you want to enter point (4.2, 8.5), then on one line in the text box used for data entry write ‘4.2, 8.5;’ without the quotes. That is, type in four point two comma eight point five semicolon. The comma between coordinates and the semicolon at the end are important. Put one data point on one line in the text box.

• •

You can graph up to 100 points. Round your minimums and maximums so they easily extend over the range of your data. The difference between each minimum and its corresponding maximum should be evenly divisible by 5 for the graph tic mark values on the axes to look good. • This graph maker works with a fixed width text font, and therefore the placement of the points is not perfect; it is rounded to the nearest text row and column. The graphs are hardly useless, however. You will get a good idea of how your data looks, and the graphs are meant to be easy to print on almost all equipment. • If you want to see how the whole thing works, just click the ‘Make graph’ button. The grapher defaults to a simple position vs. time, or x vs. t, graph. 45-45-90 and 30-60-90 Triangles This page summarizes two types of right triangles which often appear in the study of mathematics and physics. One of these right triangles is named a 45-45-90 triangle, where the angles in the triangle are 45 degrees, 45 degrees, and 90 degrees. This is an isosceles right triangle. The other triangle is named a 30-60-90 triangle, where the angles in the triangle are 30 degrees, 60 degrees, and 90 degrees. Common examples for the lengths of the sides are shown for each below.

The 45-45-90 Triangle

Here we check the above values using the Pythagorean theorem. The length of the hypotenuse should be equal to the square root of the sum of the squares of the legs of the triangle.

Listed below are the values shown in the diagram as well as another common set of values for this triangle. Be sure to notice that the two legs have the same length, so, the leg length is listed only once.

Hypotenuse Leg Length Length 1

0.7071

1.4142

1

The 30-60-90 Triangle

Here we check the above values using the Pythagorean theorem. The length of the hypotenuse should be equal to the square root of the sum of the squares of the legs of the triangle.

Listed below are the values shown in the diagram as well as another common set of values for this triangle.

Hypotenuse Leg Length Leg Length Length Opposite 30 Deg Opposite 60 Deg 1

0.5000

0.8660

2

1

1.7320

1.1547

0.5773

1

Interval Notation Interval notation is a method of writing down a set of numbers. Usually, this is used to describe a certain span or group of spans of numbers along a axis, such as an x-axis. However, this notation can be used to describe any group of numbers. For example, consider the set of numbers that are all greater than 5. If we were to write an inequality for this set, letting x be any number in the group, we would say:

This same set could be described in another type of notation called interval notation. In that notation the group of numbers would be written as:

Here is how to interpret this notation: • •

•

•

The span of numbers included in the group is often imagined as being on a number line, usually the x-axis. The ‘(5’ on the left means the set of numbers starts at the real number which is immediately to the right of 5 on the number line. It means you should imagine a number the tinniest bit greater than 5, and that is where the group of numbers begins. The parenthesis to the left of 5 is called a round bracket or an exclusive bracket. That is, 5 is excluded from the group, but the numbers directly to the right of 5 are included. Simply put, numbers greater than 5 are included. The group of numbers continues to include values greater than 5 all the way to a value which is infinitely greater than 5. That is, the set of numbers goes all the way to positive infinity. That is what the positive infinity symbol on the right means. Infinity symbols are always accompanied by round brackets.

Now consider the group of numbers that are equal to 5 or greater than 5. That group would be described by this inequality:

In interval notation this set of numbers would look like this:

This interval notation would be interpreted just like the interval above, except:

•

The ‘[5’ on the left means the set of numbers starts on the number line with 5. The square bracket to the left of 5 is called an inclusive bracket. That is, 5 is included within the group. Simply put, the number 5 and all numbers greater than 5 are included.

Now, what about numbers greater than 5 but less than 7? Expressed as an inequality this group would look like this:

This same group of numbers expressed with interval notation would look like this:

Again the round, exclusive brackets on the left and right mean ‘up to but not including’. And here is an inequality showing a group of numbers equal to or greater than 5 and less than 7:

Here is this group of numbers expressed with interval notation:

Notice that there is a square, or inclusive, bracket on the left of this interval notation next to the 5. This means that this group of numbers starts at 5 and continues for values greater than 5. The round bracket on the right next to the 7 is, again, an exclusive bracket. This means that the numbers in this group have values up to but not including the 7. Well, by now, hopefully interval notation is clear to you. Let us go through one last simple example. Consider the group of numbers equal to or greater than 5 and less than or equal to 7. An inequality for this set would look like this;

Since both the 5 and the 7 are included in the group we will need inclusive, or square, brackets at each end of the interval notation. That notation looks like this:

Well, let us get just a bit more complicated. Using interval notation we will show the set of number that includes all real numbers except 5. First, stated as inequalities this group looks like this:

The statement using the inequalities above joined by the word or means that x is a number in the set we just described, and that you will find that number somewhere less than 5 or somewhere greater than 5 on the number line. In interval notation a logically equivalent statement does not use the word or, but rather a symbol for what is called the union of two groups of numbers. The symbol for union coincidentally looks like a U, the first letter of union. However, it is really not a letter of the alphabet. Here is what the union symbol looks like:

So, the group of numbers that includes all values less than 5 and all values greater than 5, but does not include 5 itself, expressed as interval notation looks like this:

Let us consider one last set of numbers. We will consider a group of numbers containing all numbers less than or equal to 5 and also those numbers that are greater than 7 but less than or equal to 12. Using inequalities this group of numbers could be notated like this:

And using interval notation as described throughout this material this group would look like this:

We would interpret this interval notation as representing the total group of numbers as the union of two other groups. The first would start at negative infinity and proceed toward the right down the number line up to and including 5. The second would start just to the right of 7, but not including 7, and continue to the right down the number line up to and including 12. The total set of numbers would be all those in the first group along with all of those in the second, and this would be the same total group of numbers which we considered in the above inequality where we first introduced this last example.

So, we see that interval notation is useful for stating the members of groups of numbers. It is often used to state the set of numbers which make up the domain and range of a function.

What is a perfect square? A perfect square is a quadratic expression that factors into two identical binomials. For example, if you multiply these binomials: (x + 2)(x + 2) You get: x2 + 4x + 4 Therefore, the quadratic expression x2 + 4x + 4 is a perfect square since it factors into two identical binomials which are (x + 2) and (x + 2). Notice that (x + 2)(x + 2) can be written (x + 2)2. So: x2 + 4x + 4 = (x + 2)2 Here are some more examples of perfect squares: x2 + 10x + 25 = (x + 5)(x + 5) = (x + 5)2 x2 - 6x + 9 = (x - 3)(x - 3) = (x - 3)2

Where does the quadratic formula come from? First of all, let us be sure we understand what is meant by the term quadratic formula. To do that, we must understand what is meant by the terms quadratic expression and quadratic equation. A quadratic expression is a polynomial of this form:

A quadratic equation is an equation of the form:

Examples of quadratic equations would be: 0 = 6x2 + 2x + 9 0 = 3.2x2 + 8.2x - 1.6 When equations of this form are solved for x, you get you get the quadratic formula, which is:

That is, you can use this calculation to find the values of x that make the quadratic equation equal to zero. These specific values of x are called the roots of the quadratic equation. They are also called the zeros of the equation. Let us figure out why x is equal to the above value. First, start with the quadratic formula:

[1] Subtract c from both sides to get:

[2] Divide both sides by a:

[3]

This next step may look a bit mysterious at first. It will allow us to turn the left side of the above equation into a perfect square. This method of turning one side of an equation into a perfect square is often called 'completing the square'.

Add

to both sides:

[4] Factor the left side of this equation by asking this question: What two numbers

when added are

That number is

and when multiplied are

?

.

Therefore, we can see the left side of [4] as a perfect square that factors this way:

[5] The left side can now be written this way:

[6] Let us do a bit of algebra on the right side of [6]:

[7] Now, we get a common denominator on the right side:

[8] Adding on the right side:

[9] Taking the square root of each side:

[10]

Subtract

from each side:

[11] Do some algebra on the right side:

[12]

Since the terms on the right side now have a common denominator, we will combine those terms by adding them:

[13] And we are all done. [13] is the solution to the quadratic equation.

Infinity and Its Symbol In mathematics an unimaginably large value is said to be at infinity or is said to be infinity. For the moment, let us consider such a value to be positive. That is, think of an x number line with the origin, or zero, on it. In your imagination have it extend from zero to the right forever. One could say that infinity is at the end of that line. Except that line has no end. It goes on without end. So, how far away to the right of zero is positive infinity? Well, if you started at the origin and spent your whole life running toward infinity at a million miles per hour, and somehow miraculously lived for a million lifetimes, at the end of all of that you would be no closer to infinity than when you started. That puts infinity very far away indeed. The symbol for this value looks like this:

This symbol is often read ‘positive infinity’. Just like for normal numbers, the positive sign on infinity can be implied or ‘invisible’. Therefore, this symbol has the same meaning:

What about the symbol for negative infinity, that value which is forever distant to the left of the origin? That symbol looks like this:

One could think of the positive infinity symbol as a variable with a value that is larger than any other positive number or that is less, in the case of negative infinity, than any negative number.

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Table of Contents 1. How to Read Expression Evaluation 2. Introduction to Expressions 3. Introduction to Operators and Operands 4. Unary Operators 5. Raise to a Power Operations 6. Parentheses 7. Variables and Functions 8. A Few Notes About Multiplication 9. Terms and Factors 10. Linear Functions 11. Polynomial Functions 12. Rational Functions 13. Exponential Functions 14. Trigonometry Realms 15. Intersections 16. The Geometry Section 17. Simple Data Grapher 18. Interval Notation 19. What is a Perfect Square 20. Infinity and Its Symbol

Expression Evaluation

Understanding mathematics notation is, well, essential to understanding mathematics. The material here discusses what we mean by the concept of an expression, and how one reads an expression in order to evaluate it. We will center on the concepts of operators and operands and look at expressions as if they were animated in our imagination. A few Flash animations are presented in an effort to get these ideas across. Precedence is covered here, and usually students learn that quite early in their study of mathematics. Often, though, a person is not exposed to the concept of interpreting a math expression in terms of its operators and operands. At any rate, it is suggested that you start with the first link and work your way down through the list of others. Introduction to Expressions Other Representations Besides Numbers for Values Introduction to Operators and Operands More about Operators and Operands Unary Operations Raise to a Power Operation Parentheses And Now Including Variables and Functions Just a Few Notes about Multiplication

Terms and Factors

Introduction to Expressions In a very simple manner, this certainly seems true:

As many of these circles

equals

as many of these rectangles.

So, how do we talk about and write about ‘as many of’ or ‘how many’?

One way to do it is with words. Using words, here’s how many circles there are above:

Three Mathematicians usually do not use words for these types of situations, however. They prefer other ways. Another way to describe how many circles we have here is to use digits. Here’s how many circles using digits:

3

When digits are used to describe ‘how many’ we often say we are dealing with numbers. Of course, numbers can have more than one digit, and they may include a decimal point. They can also include a positive or a negative sign, but in some discussions those are not considered part of the number, but are considered an operator on the number. Now, ‘three’ or ‘3’ is said to be the value a mathematical expression that represents ‘how many’ of the circles, (or rectangles, for that matter), that we have above. That mathematical expression, again, looks like this:

3 This mathematical expression is made up of one mathematical element, the number 3. This mathematical expression evaluates to a value of ‘three’, (if a word is used), or ‘3’, (if a number is used). Mathematical expressions can have more than one number in them. They can also contain elements that are not numbers. For example, here is another mathematical expression:

3+2 This mathematical expression evaluates to a value of ‘five’ or ‘5’. Here is another mathematical expression that evaluates to a value of ‘five’ or ‘5’:

5 Since both of these two newest expressions have the same value, they are equal expressions, and we can write:

3+2=5 Or:

5=3+2 One good way to think about a value is that it is a location on a number line. A number line is a line, drawn or imagined, with values strung along it. The value at any point can be indicated by a number called the coordinate of that point. As you move to the right along the number line the values get greater. They get less and less as you move to the left. A number line has a place called the origin; this is for the value of zero. Numbers to the right of zero are positive, numbers to the left of it negative. Here’s a picture of a number line:

So, when two expressions are equal, then each of their values is located at the same spot on the number line. Now, two expressions do not have to be equal. They could be unequal. In such a case one would find that after evaluating each expression separately to its corresponding value, then those two values would be located at different spots on the number line. So, one expression can be greater in value or lesser in value than another, but for now, let’s just deal with two expressions that are equal to each other. Other Representations Besides Numbers for Values

Now, let’s just say that there are situations where one knows that a value exists, but one does not know exactly what that value is. For example, suppose you know that there are some salt shakers in this box:

How many salt shakers that are in that box is a value. (By the way, there may be zero salt shakers in that box.) That is, somewhere along the number line there is a spot that indicates the value for the number of salt shakers. But we don’t know where that spot is. However, none of this stops us from doing mathematics with the value of the number of salt shakers in this box. We just need to understand how to represent that value. Symbols called variables are used to represent the concept of ‘how many’ when that concept is considered to exist, although the exact value of that concept may be unknown at the present, (and may continue to be unknown). Well, let’s just say that variables act like numbers, but you may not know which number a variable is acting like. Variables usually look like letters of the alphabet. So, if we want to do mathematics with the number of salt shakers in that box above, we need to decide upon a letter of the alphabet to represent the ‘how many’ for that value. Let’s let the letter ‘n’ be that letter. Then we would say: Let n be the number of salt shakers in the box. If we wish to consider that there are three salt shakers in the box, then we would make statements like:

Let n equal 3. Or: Let n = 3. Or simply: n=3 It is important to know, though, that we can do mathematics with the variable n without ever knowing how many salt shakers are in the box. For example, if the box doesn’t weigh too much, it’s possible to show that the total weight of it and its contents about

doubles if the number of salt shakers inside of it doubles. One could find out that fact mathematically without ever knowing how many salt shakers were actually in the box. So, values can be represented by numbers and variables. Numbers are made up of digits and maybe a decimal point. Sometimes one considers a positive or negative sign part of a number. The value of a number is clearly represented by itself. Its location on the number line is not up to question. Variables are symbols which are usually single letters of the alphabet. Variables are often called ‘unknowns’ because the exact value for a variable may be unknown in the problem you are facing. In fact, knowing the exact value of a variable may not actually enter into the work you are doing involving that variable. For example, this is true:

5=3+2 And this is true, too:

5+n=3+2+n The above statement is true no matter what the value for n may actually be. Here we have done some algebra using a symbol, n, for a value that is unknown. So, we now have two methods for showing a value in a mathematics expression: • Use a number, say, like 3. • Use a variable, for example, n. There is another manner in which a value can be introduced into a mathematics expression: the concept of a function. Currently, we will not go into depth regarding the definition of a function. As far as notation and algebra go, however, let’s say that functions work in mathematics expressions much like a number or a variable. The symbol for a function can have slightly different meanings in different contexts. Here, let’s say that it is a symbol which represents a value. A symbol for a function often looks like this: f(x) The above symbol is pronounced like this: ‘f of x’, (‘ef of ex’) This entire symbol, f(x), acts like a value. That is, since this is true:

5=3+2 Then so is this true: 5 + f(x) = 3 + 2 + f(x) What value does this symbol, f(x), represent? That is a good question! It’s value can change, so it acts a bit like a variable. Actually, though, its value, called its output, is

determined by what is known as its input. And the rule that ties the value of its output to the value of its input is called the definition of the function. In the symbol f(x) the x stands for the input value. The name of the function is f. And the entire symbol, f(x), stands for the output value, that is, the value that enters the mathematics expression. If we know the function definition, then we can determine the output value once we know the input value. For example, suppose the function is defined as: f(x) = 5x This states that the function will have a value, or output, equal to five times the input value. For example if 2 is input to this function, it will output 10. Starting again with the definition and placing a 2 in for the x, the math looks like this: f(x) = 5x f(2) = 5(2) f(2) = 10 For the above we would say:

‘f of two equals ten’

Of course, a different input would yield a different output. For example: f(6) = 30 For the above we would say:

‘f of six equals thirty’ Now, there is a lot more to understanding functions than this. They are mentioned here because it is important to understand the common ways in which values enter mathematics expressions right from the start when working with expression evaluation. So, now we know that there are three common ways that values are represented in mathematics expressions: •

Numbers

• •

Variables Functions

Introduction to Operators and Operands As we have seen, a mathematics expression can be made up of only one element, for example, a number, as in:

3 The above expression evaluates to three. This value of three can be represented by the number 3.

Usually, though, expressions are much more involved than the simple one above. Let's consider this one:

3+2 The above expression evaluates to five. The above expression has three elements in it: • • •

The number 3 The addition operator '+' The number 2

Well, we have already dealt with the idea that 2 and 3 are numbers representing values in an expression. Here, let's talk about the '+'. Officially, the '+' in the above expression is called an operator. It is the addition operator. Operators do not exist alone in expressions. That is, for example, the addition operator needs some values to add. It needs some values to operate upon. In this expression:

3+2 The addition operator works with, or operates upon, the values represented by the numbers 3 and 2. The numbers 3 and 2 are said to be the operands for the '+', that is, the 3 and the 2 are operands for the addition operator. It is very handy to think of the addition operator as 'grabbing' the 3 and the 2 and, through arithmetic, producing a value of five.

Speaking a bit more technically, we would say that the above expression evaluates to five because the addition operator operates on the 3 and the 2, (the number 3 and the number 2 are its operands), to produce a value of five. So, now we know: • •

Mathematics expressions are made up of operators and operands. When an expression is evaluated operators accept operands, (which are values), and produce new values through arithmetic.

Specifically, regarding the addition operator we see: • • •

The symbol for the addition operator is '+'. The addition operator needs two operands, one to its left and one to its right. The operands for the addition operator are values.

Since the addition operator requires two operands, we say that it is a binary operator. Binary means 'consisting of two parts'.

There are other binary operators present in expressions. There is also the subtraction operator, as in:

7-4 Here, the subtraction operator, '-', accepts the 7 and the 4, and using arithmetic it produces a value of three.

So, we now have two examples of binary operators in mathematics expressions: • •

The addition operator, '+' The subtraction operator, '-'

Now, an expression can have several addition and subtraction operators in it. For example:

7-4+2 The operators are going to accept values as operands, but in this case, which operator gets the first turn at accepting these values, the '-' or the '+'? To answer that question, it would be handy to know which operator is 'stronger'. In mathematics, when one operator is 'stronger' than another, we say that it has precedence over the other. That is, it precedes the other in the order of the steps taken necessary to evaluate the expression. As it turns out the addition operator and the subtraction operator have equal precedence, one is not 'stronger' than the other. In cases like this, when two operators are of equal precedence, evaluation proceeds from left to right. So, in this case the subtraction operator goes first since it is left most. In detail, here is how this evaluation unfolds, starting with the original expression:

7-4+2

The '-' operator goes first. It accepts the value of seven represented by the number 7 and the value of four represented by the number 4, does arithmetic called subtraction, and produces a value of three. Since the value of three can be represented by the number 3, this work effectively changes the above expression to look like this:

3+2 Now, the action is handed over to the '+' operator. It accepts the value of three, (which was handed to it by the '-' operator), and the value of two represented by the number 2, does arithmetic called addition, and produces a value of five. The value of five can be represented by the number 5, so the above expression boils down to this one:

5

At this point we know: • • • • •

In math expressions there are operators and operands. It is the interaction among operators and operands that is called the evaluation of the expression. The evaluation of a mathematics expression produces a final value. Binary operators require two operands. The addition operator, '+', and the subtraction operator, '-', are both binary operators.

• •

The addition operator and the subtraction operator have equal precedence. When two operators have equal precedence, the one to the left accepts operands first, producing a value which is passed over to the other operator.

More About Operators and Operands Now, let's consider this situation:

2*3 The asterisk, '*', in the above equation is meant to show multiplication. So, the above reads 'two times three'. Therefore, in this example the operation of multiplication is shown by '*'. And we say that the symbol '*' is the multiplication operator. There are other ways to show multiplication. An 'X' is often used, and sometimes a dot is used, and sometimes just placing two variables next to each other means multiplication. But for now, let's just use an asterisk to designate multiplication. Notice that multiplication is a binary operation. That is, the multiplication operator, '*', is a binary operator. It accepts two operands. In the above example the two operands are the number 2 and the number 3. The multiplication operator above accepts the two operands, performs multiplication arithmetic with them, and produced the value six. The value six can be represented by the number 6, so, we can write:

6=2*3

The division operator, '/', is also a binary operator. Consider this expression:

10 / 5 This would read 'ten divided by five'. The division operator here accepts two operands, the number 10 and the number 5, and divides the left one by the right one. This produces a value of two. The value of two can be represented by the number 2, so we can write:

2 = 10 / 5 Division and multiplication have equal precedence. Consider this expression:

10 / 5 * 3 As with the similar case involving addition and subtraction, here we proceed from left to right when operators have identical precedence. So, the ten is divided by five. This produces a value of two. This value of two is next multiplied by the value of three to produce a value of six. The value of six can be represented by the number 6, so we have:

6 = 10 / 5 * 3

Here's what we have so far: • • •

Multiplication and division are both binary operators. Multiplication and division have equal precedence. Again, when operators have equal precedence, evaluation happens from left to right.

Now, understand that multiplication and division have precedence over addition and subtraction. Consider this expression:

3+2*4 The multiplication operator, '*', accepts operands, the numbers 2 and 4, before the addition operator, '+'. Therefore, two times four produces a value of eight; three added to eight produces a final value of eleven. This evaluation can be viewed this way:

3+2*4 3+8 11 So, we can write:

11 = 3 + 2 * 4 Here are some more examples to show how multiplication, division, addition, and subtraction interact. Consider:

3+8/4-1 Here, division has precedence over addition and subtraction. So, it accepts operands first. Eight divided by four is two, effectively changing the expression to this:

3+2-1 Now, addition and subtraction have equal precedence. Addition is to the left, so it accepts operands first. Three added to two equals five. So, effectively we now have:

5-1 Finally, the subtraction operator accepts operators. Five take away one is four. So, we can say:

4=3+8/4-1

Also consider:

2 * 4 - 12 / 3 In this expression both the multiplication operator, '*', and the division operator, '/', have precedence over the subtraction operator, '-'. Both the multiplication operator and the division operator have equal precedence, so we proceed from left to right with them. That makes the

multiplication operator accept operands first. Two multiplied by four is eight. So, we effectively have:

8 - 12 / 3 Next in line is the division operator. Twelve divided by three is four, effectively giving us:

8-4 And, of course, finally, eight take away four is four. So, we say:

4 = 2 * 4 - 12 / 3

In summary, so far: • • • •

Binary operators take two operands. Addition, subtraction, multiplication, and division are all binary operations. Multiplication and division have precedence over addition and subtraction. When two operators have equal precedence, the one to the left accepts operands first.

Unary Operators Not all operators in mathematics expressions are binary operators. Some are unary operators. As you may guess, a unary operator accepts only one operand. The positive sign and the negative sign are each unary operators. They accept only one value when they do arithmetic. The positive sign looks a lot like the addition operator. For example, consider this expression:

+2 Is read ‘positive two’. It does not really mean ‘add two’. What is happening here from an expression evaluation viewpoint is that the value of two represented by the number 2 is accepted as a single operand by the unary operator, ‘+’, and that operation produces a value of positive two. But, of course, the value of two represented by the number 2 is positive to start with, so, this operation is not too exciting, (no pun intended). Obviously, ‘+2’ and ‘2’ each represent the same value. So, one can say that an ‘invisible’ positive sign can be imagined before any number, (or variable or function, for that matter). As mentioned above, the negative, (or opposite), sign is a negative operator. Consider this expression: • 3 Technically, here the negative sign operator accepts a value of three as its operand and produces a value of negative three. Obviously, ‘-3’ and ‘3’ represent different values. There are no invisible negative signs in mathematics expressions. These two unary operations, ‘+’ and ‘-‘, have precedence over all binary operations. To see this, let’s look at a situation which is often termed a ‘double negative’ or stated as ‘two negatives make a positive’. Consider this expression:

3 - -2 In this expression there are two operators that look like this: ‘-‘. They are not the same. The first one, (left most), is the binary subtraction operator. The other, (right before the 2), is the unary negative sign operator. From an expression evaluation viewpoint, things proceed as follows. The negative sign operator has precedence over the subtraction operator, so it accepts operands first. It accepts only one operand, since it is a unary operator. It accepts the value of two and produces a value of negative two. Now, the subtraction operator gets its chance to accept

values. It takes the three and from it subtracts the negative two. This produces five, represented by the number 5, so we can write:

5 = 3 - -2 Now, this expression also equals five:

3+2 Since two expressions both equaling five equal each other we can write:

3 + 2 = 3 - -2 Although the expressions to the left and right of the equal sign above produce the same value, they each have quite a different story going on regarding the interaction of operators and operands. So, the positive sign operator is not the addition operator, and the negative sign operator is not the subtraction operator. But they look so much alike! Or do they? Actually, the positive sign operator is often differentiated from the addition operator by making it a bit smaller and lifting it up a bit, as in: +

2 + +3

That would read ‘positive two added to positive three’. A similar visual difference is often used for the negative sign operator and the subtraction operator. What was written above in the ‘double negative’ discussion:

3 - -2 Is much more clearly written this way:

3 - -2 That makes it very clear that the negative sign is a different operator than the subtraction operator. This last expression would be read ‘three subtract negative two’. There is really no ‘double negative’ here. One operation is binary subtraction, the other is unary negation. The combination of these in like expressions may evaluate to a value equal to a similar looking expression involving an addition operator, and an addition operator may look like a positive sign, but it is really not the best mathematics to explain the fact that:

3 + 2 = 3 - -2 Using terms like ‘double negative’ or ‘two negatives make a positive’. So, is there ever a double negative or do two negatives ever make a positive?

Well, yes, if there really are two negative sign operators present, (and not one subtraction operator and one negative sign operator). Consider this expression: • 2 Here there are two unary negative sign operators looking around, as we might say, for operands. They have equal precedence, so, which one accepts operands first? Unlike the situation for binary operators where in situations of equal precedence evaluation proceeds from left to right, here, with unary operators of equal precedence evaluation goes from right to left. So, the value of two is accepted by the right most negative sign producing a value of negative two, or the opposite of two. This value is handed over to the left most negative sign producing the negative of negative two, or the opposite of the opposite of two, which is positive two. In this case, then, it is fine to talk about double negatives and say that two negatives produce a positive. Now, with complete clarity we can write: +

2 = --2

Collecting a few ideas about expression evaluation: • • • •

There are binary and unary operators. Unary operators have precedence over binary operators. Resolutions regarding binary operations of equal precedence proceed left to right. Resolutions regarding unary operations of equal precedence proceed right to left.

Raise to a Power Our final operator has the highest precedence, is binary, and is usually invisible. Here, we are going to discuss the raise to a power operation. Basically, a raise to a power operation looks like this:

23 The two is called the base. The 3 is called the exponent. The whole expression, that is, 23, is called the power. This power would be called ‘the third power of two’. It is often called ‘two raised to the third power’, also. Just remember that, technically, the 3 is not the power. It’s the exponent. Again, the power is 23. This power, 23, evaluates to eight because 23 means two times itself three times, that is, two times two times two. There is no visible operator. It is the positioning of the exponent, the 3 in this example, to the right and up from the base, the 2 in this example, that designates the operation.

This raise to a power operation has precedence over all the binary operators (multiplication, division, addition, subtraction) and unary operators (positive and negative signs). For example, consider this example:

4 * 23 We have two operations present here: multiplication and raise to a power. The raise to a power operation goes first. So, two raised to the third power evaluates to eight, and then four times eight is thirty-two. Therefore, we can write:

32 = 4 * 23 Here is a harder one to understand: • 42 Now, the negative sign out front must wait till the raise to a power operation is finished. So, four raised to the second power is sixteen, since four times four is sixteen. After that

evaluation the negative sign accepts the value of sixteen as an operand and produces a value of negative sixteen. Therefore, we can write: • 16 = -42 In other words, -42 does not mean negative four times negative four. That would be positive sixteen. Some calculators give this result; so, be careful and make sure that you understand how the calculator that you are using works. The exponent may be negative. Consider this expression:

4-3 The negative sign on the exponent means that this power has the value of one divided by four raised to the third power. That is:

1 / 43 = 4-3 Although the above notation is not incorrect in any way, perhaps this is more clear:

The exponent can be a fraction. Taking a root, such as a square root or a cube root, is actually the raising of a number to a fractional power. Here are some examples:

Now, if the exponent contains some arithmetic, all of that arithmetic must be done before you can clearly see with what power you are working. In other words, the exponent itself can be an expression with operators and operands. For example, consider this:

32 + 4 In the above example the exponent is the expression ‘2 + 4’, which evaluates to six. So the entire expression, 32 + 4, evaluates to the sixth power of three, or equivalently three times three times three times three times three times three, or seven hundred and twenty nine. So:

32 + 4 = 36 = 729 Here is an interesting situation:

What does that mean? Well, it means two raised to some power. But what power? Well, the exponent for two contains some arithmetic which itself contains a raise to the power operation. The exponent for two is the fourth power of three, or eighty-one. In notation that looks this way:

By the way, the eighty-first power of two is quite a large number. My calculator reads: 2,417,851,639,229,258,349,412,352. I would suspect that is correct, but I really have no common experience to check it against. Sometimes an operator is shown for the raise to the power operation. This is regularity true when such an expression must be entered into a device that does not permit superscripts, such as the graphics calculator like EZ Graph. Under such conditions a caret, or ‘^’, is used. In other words, this:

3^4 Stands for this: 34

Parentheses Basically, parentheses are used in expressions to change the natural order of precedence among the operators. For example, consider this expression:

3+4*5 Since multiplication has precedence over addition, we know that this expression will be evaluated as follows: Four times five is twenty; twenty plus three is twenty-three. Or, in notation: 3+4*5 3 + 20 23 But what if you want to add the three and the four first, and then multiply by five? How do you then set up the notation? You use parentheses, like this:

(3 + 4) * 5 One always evaluates the expression inside of parentheses first. So, for the above situation three is added to four to get seven, and then seven is multiplied with five to get thirty-five. Again, in notation: (3 + 4) * 5 7*5 35 The expression inside of the parentheses may have its own precedence for operations. For example:

(2 + 4 * 6) * 3 Would work this way: Inside the parentheses multiplication has precedence over addition, so four would be multiplied by six to get twenty-four, and then two would be added to twenty-four to get twenty-six. That is the value inside of the parentheses. This value, twenty-six would then be multiplied by three to finally get a value of seventy-eight. In notation: (2 + 4 * 6) * 3 (2 + 24) * 3 26 * 3 78 Notice, regarding an example which we have seen before: • 42 = -16 That is, the negative of the second power of four is negative sixteen. But by using parentheses we can state:

(-4)2 = 16 That is, the second power of negative four is positive sixteen, or negative four raised to the second power is positive sixteen. Here is another example from before:

Notice how parentheses changes things; here, it is the third power of two which receives its value first, making a value of eight, and then this value is raised to the fourth power:

As we saw before, 281 = 2,417,851,639,229,258,349,412,352. But 84 = 4,096. Certainly changing the natural order of the operators with parentheses changes quite a bit here. Sometimes parentheses have a use that does not look much like they are being used to change the order of operator precedence. They can be used to group terms even if this grouping does not change the order of evaluation. For example, the parentheses are not really needed here, since multiplication automatically has precedence over addition: 4 + (5 * 3) However, they do no harm, and, perhaps, the author wants to show some special feature in a problem where the five times the three forms a unit of thought currently under discussion, so, it is emphasized this way.

Including Variables and Functions If you have been following the links from top to bottom on the Expression Evaluation page, you probably noticed that variables and functions were mentioned early on, and then through most of the material only numbers were used in the examples. That is because the main topics under discussion when explaining the evaluation of expressions concern operators, operands, and precedence. The operands we used were numbers, but any symbol representing a value could have been used, and all of the conclusions would remain the same. In other words, here is an expression that has in it he numbers 4, 5, and 6, and the subtraction and division operators:

4-5/6 As stated earlier, division has precedence over subtraction, so five is divided by six, and this value is then subtracted from four. Here is a similar situation with variables and functions, rather than numbers, representing values:

x - f(x) / y The same rules of precedence apply. The output value of the function, f(x), is divided by the value of y, producing some intermediate value. This intermediate value is then subtracted from x to get the final value for the expression. Of course, you would need to know the actual values for x, f(x), and y to come up with the actual final value for this expression. The purpose here, though, is to note that the order of operations works the same for numbers, variables, and functions.

And now we will examine an interesting note about the raise to a power operation, and how it is written for numbers, variables, and functions. The proper notation for the power of a number, say the second power of 4, is:

42 The proper notation for the power of a variable, say the second power of x, is:

x2 But, the proper notation for the power of a function, say the second power of f(x), is not:

f(x)2 Correctly written, the second power of f(x), or f(x) squared, is:

f2(x) Although it is usually not seen, this would be the same as (f(x))2. So:

f2(x) = (f(x))2 Now, also note that:

4-1 means 1/41 or 1/4 And:

x-1 means 1/x1 or 1/x But:

f-1(x) does not mean 1/f(x)

The symbol f-1(x) means the inverse of the function f(x). And inverse functions really have nothing to do, (necessarily), with reciprocals. If you want to state 1/f(x) using notation similar to above, then:

(f(x))-1 means 1/f(x)

A Few Notes About Multiplication We have used an asterisk for the multiplication operator. This is not really unusual; many computer languages and graphics calculators work this way. So, a statement such as this can mean three times two:

3*2 But there are other common ways to note multiplication. Each of the following are usually accepted to mean three times two:

If you were using variables, such as p and q, multiplication could be shown in any of these ways:

In fact, in such cases involving variables you can indicate a multiplication by simply placing the variables next to each other. The following means p times q: pq The above method does not work for numbers, though. If you try to state three times two using this method you end up with:

32 Which, obviously, is thirty-two, not at all equal to six. If you want to put two numbers next to each other in order to indicate multiplication, you need to use parentheses. Any of the following correctly states three times two: 3(2) (3)2 (3)(2) And what if you want to show a number times a variable? Well, here is five times x:

5x The number comes first, and no parentheses are required. Although strictly not incorrect, the following forms are not normally used in mathematics notation: x5 (5)x 5(x) (5)(x)

Terms and Factors When discussion expression evaluation it is handy to understand what is meant in mathematics by the words ‘term’ and ‘factor’. We have looked at expressions and discussed them from the viewpoint of operators and operands. Now, we will look at an expression from the viewpoint of terms and factors. From this viewpoint we are less concerned with the interactions among the parts of the expression, but more concerned with how to generally name the parts of an expression. Terms are those parts of the expression between addition signs and subtraction signs. Consider this expression: x+y-z This expression has three terms: the variables x, y, and z. We say that the first term is x, that the second term is y, and so on.

Here is another expression:

7 - f(x) + a This expression also has three terms: the number 7, the function f(x), and the variable a. The first term is the number 7, the second term is the function f(x), and the third term is the variable a. Now, consider this expression: xy + 2 Here, the first term is this: xy Notice that this term is made up of two parts, the variable x and the variable y.

Factors are the separate parts of a multiplication. Consider this expression: xy The above, of course, is a multiplication of x times y. The factors in this multiplication are x and y. In the following expression the factors are 8, p, and q:

8pq

Now, back to this expression: xy + 2 Here the first term is xy, and that first term is made up of two factors, x and y. So, terms are groups of factors. The second term in this expression is the number 2. This term is made up of one factor, 2. Consider this expression: cd2 This expression has one term which is made up of two factors. The first factor is c. The second factor is d2. In other words, powers are considered single factors. The factor d2 is not considered two factors even though it is equal to d times d. What about division? Well, it can defined in terms of multiplication, as in:

So, one could say that the factors of a/b are a and 1/b.

And clearly see that the factors of a/b are a and b-1. So, terms are inside of factors, but the factors themselves can be made up of terms which can have factors inside of them, and so it. It can get quite complicated! Consider this: (a + b + c)(d - ef) From the outer most viewpoint, this is an expression with one term, the entire expression. From this outer most viewpoint the first and only term in this expression is made up of two factors. The first factor is: (a + b + c) The second factor of this first and only term is: (d - ef) Now, let’s look at that first factor of the first term. Again it is: (a + b + c) Or simply: a+b+c This factor is itself made up of three terms: a, b, and c. Each of these terms has but one factor since each is but one variable. Let’s also look at the second factor of the first term. Again it is: (d - ef) Or simply: d - ef This factor is made up of two terms, d and ef. The first term here, d, is made up of one factor, the single variable d. The second term here is made up of two factors, e and f. So, considering the original expression: (a + b + c)(d - ef) One could correctly say that the second term of the second factor is ef. Taking it just a bit further, the second factor of the second term of the second factor of the original expression is f.

Linear Functions Linear functions are functions that have x as the input variable, and x is raised only to the first power. Such functions look like the ones in the above graphic. Notice that x is raised to the power of 1 in each equation. Functions such as these yield graphs that are straight lines, and, thus, the name linear. Linear functions come in three main forms. Slope-Intercept Form Point-Slope Form General Form Equation of a Line - Slope-Intercept Form y = mx + b Above is a program that will help you visualize how changing the values for the slope, m, and the y-intercept, b, will affect the graph of the equation y = mx + b. At first the program will be automatically cycling through several values for m and b. If you want to use the sliders to control it yourself, just press the ‘You Control’ button. Notice that when the slope, m, is positive, the line slants upward to the right. The more positive m is, the steeper the line will slant upward to the right. When the slope is negative, the line slants downward to the right, and, as the slope becomes more and more negative, the line will slant downward steeper and steeper to the right. Also, notice that when the y-intercept, b, is positive, the line crosses the y-axes above y = 0. When b is negative, the line crosses the y-axis somewhere below y = 0. In fact, b is the value on the y-axis where the line passes through this axis. The line intercepts, or crosses, the y-axis here, and, therefore, b is called the y-intercept.

Summary of Details This linear function:

f(x) = mx + b May be graphed on the x, y plane as this equation: y = mx + b • This equation is called the slope-intercept form for a line. • The graph of this equation is a straight line. • The slope of the line is m. • The line crosses the y-axis at b. • The point where the line crosses the y-axis is called the y-intercept. • The x, y coordinates for the y-intercept are (0, b). Equation of a Line - Point-Slope Form y = m(x - x 1 ) + y 1

or

y - y 1 = m(x - x 1 )

Above is a program that will help you visualize how changing the values for the point, (x1, y1), and for the slope, m, will affect the graph of the equation y = m(x - x1) + y1. At first the program will be automatically cycling through several values for x1, y1, and m. If you want to use the sliders to control the program yourself, just press the ‘You Control’ button. Notice the point, shown as a little square, which moves around the graph. That point is (x1, y1). The point-slope form equation gets its name from the fact that if you know one point on the line, (x1, y1), and the slope of the line, m, then you can determine, or draw, the line completely. Experiment with controlling the program yourself. You should be able to realize that any line which you can imagine on the x, y plane can be drawn knowing only one point on the line and the slope of the line.

Summary of Details This linear function: f(x) = m(x - x1) + y1 May be graphed on the x, y plane as this equation: y = m(x - x1) + y1 This equation is often also written as: y - y1 = m(x - x1) • This equation is called the point-slope form for a line. • The graph of this equation is a straight line. • A known point on the line is (x1, y1). • The slope of the line is m. Equation of a Line - General Form

Ax + By + C = 0

or

y = (-A/B)x + (-C/B)

Above is a program that will help you visualize how changing the values for the variables A, B, and C will affect the graph of the equation y = (-A/B)x + (-C/B). At first the program will be automatically cycling through several values for A, B, and C. If you want to use the sliders to control it yourself, just press the ‘You Control’ button. Notice that the slope is equal to the opposite of A divided by B, that is, (-A/B). Also, notice that the y-intercept is equal to the opposite of C divided by B, that is, (-C/B).

Summary of Details This linear function: f(x) = (-A/B)x + (-C/B) May be graphed on the x, y plane as this equation: y = (-A/B)x + (-C/B) This equation is often also written as:

Ax + By + C = 0 • • • •

This equation is called the general form for a line. The graph of this equation is a straight line. The slope of the line is (-A/B). The y-intercept is (-C/B).

Polynomial Functions Stated quite simply, polynomial functions are functions with x as an input variable, made up of several terms, each term is made up of two factors, the first being a real number coefficient, and the second being x raised to some non-negative integer power. Actually, it’s a bit more complicated than that. Definition of a Polynomial Function Here a few examples of polynomial functions: f(x) = 4x3 + 8x2 + 2x + 3 g(x) = 2.5x5 + 5.2x2 + 7 h(x) = 3x2 i(x) = 22.6 Polynomial functions are functions that have this form: f(x) = anxn + an-1xn-1 + ... + a1x + a0 The value of n must be an nonnegative integer. That is, it must be whole wsnumber; it is equal to zero or a positive integer. The coefficients, as they are called, are an, an-1, ..., a1, a0. These are real numbers. The degree of the polynomial function is the highest value for n where an is not equal to 0. So, the degree of g(x) = 2.5x5 + 5.2x2 + 7 is 5. Notice that the second to the last term in this form actually has x raised to an exponent of 1, as in: f(x) = anxn + an-1xn-1 + ... + a1x1 + a0 Of course, usually we do not show exponents of 1. Notice that the last term in this form actually has x raised to an exponent of 0, as in:

f(x) = anxn + an-1xn-1 + ... + a1x + a0x0 Of course, x raised to a power of 0 makes it equal to 1, and we usually do not show multiplications by 1. So, in its most formal presentation, one could show the form of a polynomial function as: f(x) = anxn + an-1xn-1 + ... + a1x1 + a0x0 Here are some polynomial functions; notice that the coefficients can be positive or negative real numbers. f(x) = 2.4x5 + 1.7x2 - 5.6x + 8.1 f(x) = 4x3 + 5.6x f(x) = 3.7x3 - 9.2x2 + 0.1x - 5.2 Roots of Polynomial Functions

Real roots Synthetic Division Real Roots of Polynomial Functions Definition of terms and symbols when dividing polynomials:

Dividend:

f(x)

Divisor:

h(x)

Quotient:

q(x)

Remainder: r(x) If any of these are constants, for example if r(x) is constant, as in:

r(x) = 5 or: r(x) = a then variable, rather than function, notation may be used for that value, as in: r=5 or: r=a When f(x) is divided by h(x), the result is the value of q(x) plus r(x), as in: f(x)/h(x) = q(x) + r(x) This can also be written as: f(x) = h(x)q(x) + r(x)

The remainder, r(x), will either be equal to 0, or it will be less in degree than the degree of the divisor, h(x). If h(x) has a degree of 1, then the degree of the remainder must be 0. That is, the remainder must be a constant, as in: r(x) = cx0 = c Under these conditions variable notation is fine, as in: r=c Therefore, if f(x) is divided by the linear polynomial (x - c), the remainder is a constant, r. Again, consider our basic definition of polynomial division: Dividend: f(x) Divisor: h(x) Quotient: q(x) Remainder: r(x) f(x) = h(x)q(x) + r(x) Make the divisor, h(x), equal to the zero degree polynomial (x - c). This will create a remainder, r, that is a constant. h(x) = (x - c) Then this: f(x) = h(x)q(x) + r(x) Becomes: f(x) = (x - c)q(x) + r Let x = c, so: f© = (c - c)q(x) + r f© = (0)q(x) + r f© = 0 + r f© = r Therefore, when the polynomial function, f(x), is divided by a linear polynomial function in the form (x - c), the remainder is f©. This we will call the remainder theorem for polynomial division. Using this remainder theorem, if the divisor is the linear function (x - c) as in: h(x) = (x - c) Then our basic definition of polynomial division: f(x) = h(x)q(x) + r(x) Becomes: f(x) = (x - c)q(x) + f©

Suppose c is a zero of f(x). Then, of course: f© = 0 And the above: f(x) = (x - c)q(x) + f© Would become: f(x) = (x - c)q(x) + 0 Or: f(x) = (x - c)q(x) That clearly makes (x - c) a factor of f(x). Therefore, if c is a zero of f(x), then (x - c) is a factor of f(x). That means that f(x) can be though of as the product of (x - c) times some other polynomial function, q(x). This could be summarized as: If f© = 0 Then f(x) = (x - c)q(x) So, whenever we know a root, or zero, of a function, we know a factor of that function. Now we are in a position to understand a method for analytically solving a certain group of problems regarding finding roots of polynomial functions. Suppose you have a polynomial function of degree 3, and you wish to find the real, possibly integer, roots. This function might look like: f(x) = x3 - 9x2 + 26x - 24 Now, suppose you knew one root for this function. One happens to be x = 2. Check: f(2) = 23 - 9(22) + 26(2) - 24 f(2) = 8 - 36 + 52 - 24 f(2) = 0 That means: f(x) = (x - 2)q(x) Or: x3 - 9x2 + 26x - 24 = (x - 2)q(x) We can find q(x) by dividing f(x) by (x - 2), as in: q(x) = (x3 - 9x2 + 26x - 24) / (x - 2) This yields: q(x) = x2 - 7x + 12 Which means that this:

f(x) = (x - 2)q(x) Becomes: f(x) = (x - 2)(x2 - 7x + 12) Now, the second factor, (x2 - 7x + 12), is easily factored, as: (x2 - 7x + 12) = (x - 3)(x - 4) So, this: f(x) = (x - 2)(x2 - 7x + 12) Becomes: f(x) = (x - 2)(x - 3)(x - 4) Therefore we have factored f(x) into a group of linear factors. The roots of f(x) are clearly demonstrated. They are 2, 3, and 4. So, if we know one root of a cubic polynomial function, we know a linear factor of that function. If we know a linear factor of the cubic we can divide the cubic by that factor and get another polynomial factor of one degree less than the cubic. That is, we will get a quadratic polynomial function as a quotient. That quadratic can be factored by convention means, certainly by the quadratic equation. This factored quadratic equation will yield the other linear factors of the original cubic. So, a cubic can be factored into a group of linear factors using these methods. When it is so factored, its roots are obvious. This method can be generalized to situations concerning finding the roots of polynomial functions with degrees past the third. For example, if you had a polynomial function of the fourth degree, and if you could find one zero of that function, then you would have one linear factor of that function. You could divide the fourth degree polynomial by this linear function and find a polynomial of degree three. This third degree polynomial function could be factored as described above. At the moment there are some things to be concerned with: The above methods depend upon knowing at least one zero of the polynomial function so as to generate that first linear factor which will subsequently be used as a divisor to reduce the the degree of the original polynomial function. How does one find this zero? Once the process has gone on long enough, and enough linear functions have been found and used as divisors until the current quotient is of degree two and can easily be factored using the quadratic equation, what does it mean if this quadratic function factors into less than two linear factors, say one or none?

Synthetic Division Consider this polynomial function:

f(x) = 4x3 - 3x2 + x - 4 Suppose that we evaluate it at an input of x = 2, like this: f(2) = 4(23) - 3(22) + (2) - 4 f(2) = 32 - 12 = 2 - 4 f(2) = 18 In this process we raised the input to a power, as in 23. Let us see that there is a way to evaluate this polynomial function using only multiplication and addition. Start with the original polynomial and factor out an x. So, this: 4x3 - 3x2 + x - 4 Becomes: x(4x2 - 3x + 1) - 4 Factor out another x from the parenthesized expression: x(x(4x - 3) + 1) - 4 Now, imagine that you evaluate f(x) at x = 2. Begin with the inner most expression. Place a 2 for the input value of x, as in: x(x(4(2) - 3) + 1) - 4 Now you would multiply 2 (the input) by 4 (the original coefficient of x3) and then add -3 (the original coefficient of x2). This would evaluate to 5. The expression now looks like: x(x(5) + 1) - 4 Place a 2 for the next input value of x, as in: x(2(5) + 1) - 4 Now you would multiply 2 (the input) by 5 and then add 1 (the original coefficient of x). This would evaluate to 11. The expression now looks like: x(11) - 4 Place a 2 for the last input value of x, as in:

2(11) - 4 Now you would multiply 2 (the input) by 11 and then add -4 (the original final constant of the polynomial). This would evaluate to 18, the same value which we calculated at first, of course. Now, this method of evaluating the output value of a polynomial can be captured in a type of short-hand notation by positioning the input value and the coefficients of the polynomial in a certain form. Again, consider this polynomial: f(x) = 4x3 - 3x2 + x - 4 We will picture it evaluated at the input value x = 2. Arrange the input value, the coefficients, and a line like this: 2) 4 -3 1 -4

Now drop down the 4: 2) 4 -3 1 -4 4 Multiply the input 2 times the 4. Place this product, 8, under the -3:

2) 4 -3 1 -4 8 4 Add the -3 and the 8. Place this sum, 5, under the line:

2) 4 -3 1 -4 8 4 5 Multiply the input 2 times the 5. Place this product, 10, under the 1:

2) 4 -3 1 -4 8 10 4 5 Add the 1 and the 10. Place this sum, 11, under the line:

2) 4 -3 1 -4 8 10 4 5 11 Multiply the input 2 times the 11. Place this product, 22, under the -4:

2) 4 -3 1 -4 8 10 22 4 5 11 Add the -4 and the 22. Place this sum, 18, under the line:

2) 4 -3 1 -4 8 10 22 4 5 11 18

We are done. This final value, 18, is the function output for f(x) evaluated with an input of 2. This process is called synthetic division. At first this term may be confusing. What does a method of function evaluation have to do with division? Consider dividing f(x) = 4x3 - 3x2 + x - 4 by (x - 2). Standard polynomial division would look like this: 4x2 + 5x + 11

--------------------3

x - 2 )4x - 3x2 + x - 4

4x3 - 8x2 5x2 + x 5x2 - 10x 11x - 4 11x - 22 18 Compare the above with our previous synthetic division:

2) 4 -3 1 -4 8 10 22 4 5 11 18 Notice that the remainder from the standard polynomial long division is the last value calculated in the synthetic division. Notice that the coefficients of the quotient from the standard polynomial long division are aligned before the remainder on the last row, under the line, of the synthetic division. So, this synthetic division method can be used to determine the quotient and remainder when a polynomial function, f(x), is divided by a linear function of the form (x - c). Specifically, this method can be used to determine if a certain value, c, is a root of the polynomial function f(x). If c is a root, then the synthetic division method will show a zero in the remainder position. This would be equivalent to determining that (x - c) is a linear factor of f(x). An added benefit of this method is that it yields the coefficients of the quotient when f(x) is divided by (x - c). This quotient can be further examined and factored using synthetic division until all the linear factors of the original polynomial function, f(x), have been found. Once all the linear factors of the polynomial function have been found, all the roots of the function are obvious.

Polynomial Function Graphs The following are Java applets that demonstrate and animate the graphs of several polynomial functions. Zero Degree First Degree Second Degree Third Degree Fourth Degree Fifth Degree

Polynomial Function of Zero Degree A zero degree polynomial function can be defined like this:

f(x) = a 0th Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficient a. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. Usually, a zero degree polynomial function is called a constant function. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0 Polynomial Function of the First Degree

A first degree polynomial function can be defined like this:

f(x) = ax + b 1st Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a and b. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. Usually, a first degree polynomial function is called a linear function. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0

Polynomial Function of the Second Degree A second degree polynomial function can be defined like this:

f(x) = ax2 + bx + c 2nd Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a, b, and c. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. Usually, a second degree polynomial function is called a quadratic function. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0 Polynomial Function of the Third Degree

A third degree polynomial function can be defined like this:

f(x) = ax3 + bx2 + cx + d 3rd Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a, b, c, and d. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0 Polynomial Function of the Fourth Degree A fourth degree polynomial function can be defined like this:

f(x) = ax4 + bx3 + cx2 + dx + e 4th Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a, b, c, d, and e. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0 Polynomial Function of the Fifth Degree

A fifth degree polynomial function can be defined like this:

f(x) = ax5 + bx4 + cx3 + dx2 + ex + f 5th Degree Polynomial Function Demonstration This demonstration is meant to show how the shape of the graph of this function depends upon the values of its coefficients a, b, c, d, e, and f. Change these values by moving the sliders with your mouse, and notice how the this alters the form of the graph. The bounds of this graph are:

x minimum = -10.0 x maximum = 10.0 y minimum = -10.0 y maximum = 10.0

Rational Functions A rational function is formed when a polynomial is divided by another polynomial. Below are some links to sections dealing with rational functions. There is a wealth of material here that would be helpful to any student of mathematics studying this topic.

The following link takes you to an overview of rational functions. It presents a definition of a rational function as long as some introductory material concerned with finding the roots and discontinuities of rational functions.

Rational Function Definition These next links present some fundamental ideas concerning one of the simplest of rational functions, f(x) = 1/x.

A Very Simple Rational Function: f(x) = 1/x Transformations of f(x) = 1/x This next link gives a detailed explanation of how to work with a rational function. This includes a complete presentation of how to find roots, discontinuities, and end behavior.

Simple 2nd Degree / 2nd Degree The following links are all to special purpose graphing applets that each present a common rational function. These are useful to expand your understanding of the material covered in the above material. 1st Degree / 1st Degree 1st Degree / 2nd Degree 1st Degree / 3rd Degree 2nd Degree / 1st Degree 2nd Degree / 2nd Degree 2nd Degree / 3rd Degree 3rd Degree / 1st Degree

3rd Degree / 2nd Degree 3rd Degree / 3rd Degree Definition of a Rational Function A rational function is basically a division of two polynomial functions. That is, it is a polynomial divided by another polynomial. In formal notation, a rational function would be symbolized like this:

Where s(x) and t(x) are polynomial functions, and t(x) can not equal zero.

An Example Here is an example of a rational function:

To understand the behavior of a rational function it is very useful to see its polynomials in factored form. The polynomials in the numerator and the denominator of the above function would factor like this:

The Domain Now the roots of the denominator are obviously x = -3 and x = 6. That is, if x takes on either of these two values, the denominator becomes equal to zero. Since one can not divide by zero, the function is not defined for these two values of x. We say that the function is discontinuous at x = -3 and x = 6. Other values for x do not cause the function to become undefined, so, we say that the function is continuous at all other values for x. In other words, all real numbers except -3 and 6 are allowed as inputs to this function. The domain for the function, therefore, as expressed in interval notation is:

The x-intercepts The x-intercepts for this function would be where the output, or y-value, equals zero. A rational function can be considered a fraction, and a fraction is equal to zero when the numerator is equal to zero. For our rational function example this happens when the polynomial in the numerator is equal to zero, and this will happen at the roots of this polynomial. The roots of the numerator polynomial are x = -5 and x = 4. That is, when x takes on either of these two values the numerator becomes zero, and the output of the function, or y-value, also becomes zero. So, the x-intercepts for this rational function are x = -5 and x = 4. Notice that the function is defined at these two values. (It is only not defined at x = -3 and x = 6.) That makes these true x-intercepts. If the function was not defined at x = 4 because 4 was a root of the denominator polynomial, (which it is not in our example here), then x = 4 would not be an x-intercept even though it made the numerator equal to zero. One can not have an xintercept for a function at a point where the function does not exist!

The y-intercept What about the y-intercept? Well, they occur where the input, or x, value equals zero. If we look at our first un-factored form for this function, expressed in ‘y =’ form, we have:

Now, setting x = 0 we get:

That is, the graph crosses the y-axis at y = 10/9 (about 1.11). Notice that when you express the polynomials of a rational function in standard form, then the y-intercept is simply the ratio of the final terms for the two polynomials.

The Graph Let us look at a graph of this function:

The red lines are the x- and y-axes; the blue grid marks off unit distances; the green line is the function. Realize the following: •

• •

The function line is discontinuous, or ‘breaks’, at x = -3 and x = 6. That is, if you were drawing the graph by hand, you would have to lift the pen off the paper at at x = -3 and x = 6. That is what we mean by a discontinuity. Notice that these locations for the discontinuities, (x = -3 and x = 6), are the same as we reasoned above. That is, x = -3 and x = 6 are the roots of the polynomial in the denominator. The function crosses the x-axis at x = -5 and x = 4. These are the same as the values which we calculated above for the x-intercepts. The function crosses the y-axis just a bit above 1, at about 1.1. This is the same location as the calculated y-intercept above.

The End Behavior Model Function The way that a function is shaped when the input, or x, is very large positively or very large negatively is called its end behavior. That is, the end behavior of a function describes how it acts as x approaches positive and negative infinity. One might say that the end behavior of a function is described by how it ‘heads out’ to the right and left on the graph far away from the origin. In the graph above notice how the graph is leveling off to a height of about y = 1 as x gets large positively or negatively. That is, at the left and right edge of the graph the function starts to act like the function described as:

This function, y = 1, is called the end behavior model function for the rational function on the graph. You can find it without drawing the graph by dividing the leading term of the polynomial in the numerator by the leading term of the polynomial in the denominator. Our function under consideration is this:

The leading term of the polynomial in the numerator is:

The leading term of the polynomial in the denominator is:

The division of these two leading terms, numerator leading term over denominator leading term, is:

And, of course, this makes the following end behavior model function:

The Range An examination of the graph above will indicate that all real numbers are available for output. The left most fork of the graph looks like it will rise not quite to y = 1, and the right most fork looks like it may not quite drop to y = 1; so, one might at first suggest that y = 1 is not in the range. However, if you examine the central portion of the graph, you

can see that y = 1 is definitely a member of the range. So, all real numbers are present in the range. In interval notation the range would look like this:

Summary This has been an introduction to rational functions and their behaviors. For more details and examples, examine the other topics in the rational function section of the Function Institute. The link labeled ‘Simple 2nd Degree / 2nd Degree’ goes over the material presented here in much more detail and contains an animated Java applet rational function graphing utility, for example. f(x) = 1 / x This is probably the simplest of rational functions:

Here is how this function looks on a graph with an x-extent of [-10, 10] and a y-extent of [-10, 10]:

First, notice the x- and y-axes. They are drawn in red. The function, f(x) = 1 / x, is drawn in green. Also, notice the slight flaw in graphing technology which is usually seen when drawing graphs of rational functions with computers or graphic calculators. At the bottom center

of the picture you will see that the graph line appears to be heading toward the edge of the diagram, but is cut short of that. Actually, the true graph of the function continues downward past the edge of the picture. As we will shortly see, this section of the graph holds what is termed an asymptote, and computers, along with graphic calculators, often have a difficult time drawing functions near asymptotes. Notice that for this function a small positive input value yields a large positive output value. And notice that a large positive input value yields a small positive output value. Here is a picture showing that:

A complementary situation occurs for negative values. A small negative input will output a large negative value, and a large negative input will output a small negative value. Here is a picture showing this idea:

This makes complete sense if you think about it for a moment. Consider a large positive input value, say one million, i. e., 1,000,000. The output of f(x) = 1/x would be one millionth, i. e., 1/1,000,000 or 0.000001. This and other representative examples are shown in the following table:

Input value, or x

Output value, or y

1,000,000

0.000001 (or 1/1,000,000)

0.000001 (or 1/1,000,000)

1,000,000

-1,000,000

-0.000001 (or -1/1,000,000)

-0.000001 (or -1/1,000,000) -1,000,000 Let us look at this function as it leaves the graph at the top and bottom. You should notice that the green function line approaches, but does not touch, the y-axis. If you graphed the function on a set of x, y axes that went up to positive one million and down to negative one million, the function line would still not touch the y-axis, though it would get very close. Just think about the x, y values in the table above. At, say, an output value, or y value, of 1,000,000 the input would not be 0. And, of course, the input value, or x value, must be 0 for the graph to touch the y-axis. This type of behavior about the y-axis is called asymptotic behavior. And, in this case, the y-axis would be called a vertical asymptote of the function. That is, the function approaches the y-axis ever closer and closer, but never touches it. Notice that the x-axis functions as a horizontal asymptote for this function. That is, as the function line stretches out to the left or right it gets closer and closer to the x-axis, but it never touches it. So, for the function f(x) = 1/x the y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote. In the following diagram of this function the asymptotes are drawn as white lines.

The function f(x) = 1/x is an excellent starting point from which to build an understanding of rational functions in general. It is a polynomial divided by a polynomial, although both are quite simple polynomials. Be sure that you understand the concept of an asymptote, especially a vertical asymptote, and then go on to the other rational function information.

Transform of f(x) = 1 / x This Java applet demonstrates several transforms of the function f(x) = 1/x. You can assign different values to a, b, h, and k and watch how these changes affect the shape of the graph.

2nd Degree / 2nd Degree Below is a Java applet which demonstrates the behavior of a rational function. The type of rational function which it demonstrates is a second degree polynomial divided by a second degree polynomial, that is, a quadratic divided by a quadratic. Right below the applet is a very brief discussion of how to use it. That is followed by a detailed discussion of the applet and of rational function behavior in general. This applet graphs this function:

Quick instructions • • • • • • •

The graph has an x-extent of [-10, 10], and a y-extent of [-10, 10]. The x- and yaxes are drawn in red. Variables a and b are the roots of the numerator. Variables c and d are the roots of the denominator. Variables a, b, c, and d are controlled by neighboring scrollbars. Vertical asymptotes are drawn as white lines. Removable discontinuities are drawn as small white circles. Checking “Numerator” causes the polynomial in the numerator to be drawn along with the rational function. Checking “Denominator” causes the polynomial in the denominator to be drawn along with the ration function.

Detailed explanation of applet and rational function behavior in general

Roots Again, this graphing program presents the graphs of rational functions which are second degree polynomials divided by second degree polynomials. An example of such a rational function would be:

You need to visualize this rational function with the polynomials in factored form. That is, the above function, f(x), would need to be seen as:

When investigating rational functions, the first move is usually to factor the polynomials like above. You factor both polynomials to find their roots. In the above example the roots of the numerator are: roots of the numerator: x = 3 and x = -5

And the roots of the denominator are: roots of the denominator: x = 6 and x = -2 In the above applet, the variables a and b are the roots of the numerator, and c and d are the roots of the denominator. So, you should imagine rational functions of this form:

If you were to multiply the above factors in the numerator and denominator, you would get a function that looked like this:

This last rational function might look a bit more normal than its factored version if one is thinking about the division of two polynomials, since polynomials are most often viewed in standard form. However, the roots of the polynomials are at the heart of this discussion, and finding them is the first step in analyzing a rational function.

To understand rational functions it is important to understand the roots of the polynomials that make up the rational function.

Discontinuities Now, one can not divide by zero, so, when the polynomial in the denominator is equal to zero, the rational function is undefined. That is, there may be certain input values for x that cause the polynomial function in the denominator to output a zero. Consider, again, this function:

And, to clearly see the roots, perhaps it is better to view the function like this:

If x = 6, then the value of the denominator is zero since (x - 6) becomes (6 - 6), which is zero, and zero times the other factor, stated as (x + 2) or (x - (-2)), but now valued at (6 + 2) = 8, is zero. That is, zero times 8 is zero. The function is undefined at x = 6 because you can not divide by 0. For similar reasons the function will be undefined at x = -2. This value for x also makes the denominator equal to zero. At the places where the function is undefined we say that the function is discontinuous. And we say that there are discontinuities in the function at the x-coordinates where it is undefined. Therefore, in the above function there are discontinuities at x = 6 and at x = 2.

Discontinuities are locations for x values that make the function undefined.

Asymptotes These discontinuities at x = 6 and at x = -2 show up on the graph as vertical asymptotes. Not all discontinuities in rational functions create vertical asymptotes, but these do. Some discontinuities are not asymptotic, but are removable. Vertical asymptotes show up as vertical white lines on the above applet. To plug the above function into the graphing applet you must set a, b, c, and d to the following values:

a = 3 b = -5 c = 6 d = -2 That is, make the roots of the quadratic in the numerator equal to 3 and -5, and make the roots in the denominator equal to 6 and -2. Go ahead and try that. It should look like the following picture. (Do not check the “Numerator” nor “Denominator” checkbox.)

Notice that the above graph shows a vertical asymptote at x = 6. (Remember, the red lines are the x- and y-axes. Each blue line is an increment of 1.) There is also a vertical asymptote at x = -2, but let us concentrate on the one at x = 6 for the moment. The function is undefined at x = 6 because that input value makes the denominator of the function equal to 0. To the immediate left of 6 the function outputs very large negative numbers making the graph of the function drop very steeply. We could prove this behavior by finding the value of the function with an input value a tiny bit to the left of x = 6. Moving over by 0.001 usually works fine. So, let us find the output value for x = 5.999:

Clearly, to the immediate left of the asymptote the function takes on an output value much more negative than can be seen on the above picture. The above graph only goes to -10 along the y-axis. Let us see what is happening to the immediate right of the vertical asymptote at x = 6. Again, we will step over by the tiny step of 0.001. This would create an input value of 6.001. Here is the output calculation:

Well, it certainly looks like the function takes on very large positive values to the immediate right of x = 6. This behavior, large negative output to the left of x = 6 and large positive output to the right of x = 6, is what causes the asymptotic shape of the function at x = 6. It is important to remember that the vertical asymptote is not a line that is part of the graph of the function. For this reason the asymptote is drawn in a different color. It is white, the actual function is green. The white asymptote line is a visual convenience in the diagram to show a line that the function approaches, but never touches. It is a good idea to see the graph of the function without this aid. The only problem with that is that often asymptotic behavior is not completely drawn to the edges of the diagram by computers. Ultimately, this is due to the fact that the computer diagram is made up of tiny discrete dots, called pixels, while the x, y real number plane is continuous. All the same, un-check the “Asymptotes” checkbox and redraw the function. It should look like this:

Discontinuities may be asymptotic.

Discontinuities and the denominator polynomial As we have seen this function has two discontinuities at the roots of the polynomial in the denominator, x = 6 and at x = -2. Now, the denominator is a second degree polynomial, often called a quadratic polynomial. A second degree polynomial graphs as a parabola. So, if we could see the graph of the denominator polynomial, we would see a graph of a parabola. Where that parabola crosses the x-axis locates the roots, or zeroes, of that parabola. These, of course, are the locations of the asymptotic discontinuities which we have been discussing. To see both the denominator parabola and the rational function graphed together check the ‘Denominator’ checkbox. It might be useful to have the asymptotes turned on, also. Try that. Here is a diagram that shows what it should look like, along with an explanatory note:

Discontinuities are located at the roots of the denominator polynomial.

X-intercepts This function crosses the x-axis at two points. These points are called its x-intercepts. Simply put, an x-intercept will exist where the y, or output, value of the function equals zero. Now, the function we consider here is a rational function, so, since it is a division of polynomials, it may be considered a fraction. A fraction will be equal to zero only when its numerator is equal to zero. The numerator will be equal to zero at the roots of the polynomial in the numerator. The roots of the numerator polynomial are x = 3 and x = -5, as another quick look at our rational function in factored form will reveal:

So the x-intercepts should be at x = 3 and x = -5, since these make the y, or output, value of the function equal to zero. Another look at our current graph will demonstrate this:

X-intercepts are where y =0 (and the function is defined).

X-intercepts and the numerator polynomial The numerator polynomial graphs as a parabola, since it is of second degree. The roots of this parabola are the values which make the numerator of the rational function equal to zero, as explained above. On the applet above if you check the ‘Numerator’ checkbox, then the numerator polynomial will be drawn along with the rational function. You should see that the roots of this parabola are the x-intercepts discussed directly above. Try that. Probably best seen with the asymptotes turned off, it should look like the following:

The roots of the numerator polynomial are the xintercepts (if the function is defined there).

Y-intercept The y-intercept of a function occurs at the point where the function line crosses the yaxis. This would be where x = 0. So, if you set x equal to zero in the rational function, the function will output a value equal to the y-intercept. This is actually quite easy to do for rational functions. Since both the numerator and the denominator are polynomials in x, every term except the last constant term, (if present), in each polynomial becomes equal to zero. So, the value for the y-intercept is simply the ratio of the two last constant terms. This is easier to see than to explain. Here is our function again:

Or, expressed in ‘y =’ form:

Now, to find the y-intercept plug in x = 0 and solve:

Again, let us look at our graph. Notice that it crosses the y-axis at y = 1.25:

The y-intercept is the ratio of the two last constant

terms of the numerator and denominator polynomials.

End behavior model function A rational function is said to have an end behavior model function. The behavior of the end behavior model function and the behavior of the rational function are the same at large positive and large negative values of x. In other words, the rational function and the end behavior model function act the same way as they each ‘head out’ to distant left and right values along the x-axis. The end behavior model function, which we will here name m(x), is found by dividing the leading term of the numerator polynomial by the leading term of the denominator polynomial. One, of course, must view the rational function with the polynomials in standard form, not factored form. Here is another look at our function, polynomials in standard form:

For the function considered here the leading term of the numerator is x2, and the leading term of the denominator is also x2. The division of these two, as described above, yields the end behavior model function, m(x), as shown by:

Or, if you want to consider this end behavior model function in ‘y =’ form, it would look like:

This, of course, is a horizontal line that crosses the y-axis at y = 1. That means that our rational function will approach this horizontal line for large positive and large negative input, or x, values. Although the graph on the applet above does not go far enough to the left or right to see this perfectly, one can see this end behavior fairly clearly. Below is our current graph with the end behavior model function drawn in as a white line:

You should notice that for the rational functions which are presented with this applet the only possible end behavior model is m(x) = 1, since the division of the leading terms of the polynomials in the numerator and denominator will always be x2 / x2.

The end behavior model function demonstrates the behavior of the rational function at extreme values for x.

Discontinuities which are not asymptotes Go to the applet and set up the following situation. Be sure to notice that the only difference is that now x = 6 is a root of both the numerator and the denominator:

a = 6 b = -5 c = 6 d = -2 The graph should now look like this:

Notice that there used to be an asymptote at x = 6, but now there is not. (The asymptote at x = -2 is still there. We are not discussing that here.) There is still a discontinuity at x = 6, that is, the function is still not defined at there because this value for x still makes the denominator equal to zero. And still if one were to imagine drawing the function line carefully, one would have to break the function line by picking up one’s pen at x = 6. That break is represented on the graph by the little white circle. However, this other type of discontinuity is not asymptotic, it is termed removable. We say that there is a removable discontinuity at x = 6. The discontinuity at x = 6 is called removable because it can be removed, (or ‘plugged’), simply by redefining the function at this point. That is, we could assign a certain value to the output, any value, really, at x = 6. A piecewise-definition of the function would work fine, such as:

Notice that all we did was use the normal definition of our rational function for all values of x except x = 6, and, of course x = -2, which is our asymptotic discontinuity. Then we gave an output value of 1 to the function when x = 6. To remove the discontinuity it is not necessary to ‘plug’ the hole in the function line. One simply must provide an output value for the function for the value of x that causes the removable discontinuity. There is, of course, an actual (x, y) point on the real number plane that would ‘plug’ the hole, and,

perhaps, in some applications one would want to know it and use it in a similar piecewise-definition for the function. Finding that point is a calculus problem. Lastly, let us notice something about x-intercepts with this version of our rational function. Suppose we did not redefine the function so as to remove the discontinuity at x = 6. Then we are back to the function in this form:

It may look at first glance that there should be an x-intercept at x = 6 because this value makes the numerator equal to zero, which makes the output, or y, value equal to zero, and that is what we mean by an x-intercept. Except the function can not have an x-intercept at x = 6 because it is not defined there. It is discontinuous, (with a removable discontinuity), at x = 6.

Removable (and not asymptotic) discontinuities form when the the numerator and denominator polynomials share common roots. Conclusion

Well, if you got this far, you gone through most of what you need to know to understand rational functions of many types, not just second degree polynomials divided by second degree polynomials. For example, this same reasoning would be used in an investigation of a first degree polynomial divided by a third degree polynomial. Go back to the applet and play with it. Try out several combinations of for roots both in the numerator and denominator. Adjust it till you understand how to generate asymptotic and removable discontinuities. And, of course, go on to explore the other topics about rational functions in Zona Land.

P. S., So why is this called the simple version? Perhaps you remember that the link from the rational functions home page labeled this ‘Simple 2nd Degree / 2nd Degree’. Well, observant mathematicians will notice that this applet will not graph every type of rational function that is a second degree polynomial divided by a second degree polynomial. It only demonstrates those in which the leading coefficient is equal to one in both the numerator and the denominator. For example, you could not study this rational function with this applet:

This function factors like so:

The applet here provides no way to enter the factor of 3 in the numerator nor the factor of 4 in the denominator. However, the behavior of such functions as this one can be understood using the same methods described here. In fact, the behavior of most rational functions can be understood with these methods. As one closing note, see that the end behavior model function for the rational function above is not y = 1, but:

The end behavior model function is still a horizontal line. This is really the only difference one would need to consider in understanding this type of second degree polynomial over second degree polynomial rational function. Rational functions of this more general type will be demonstrated in other obvious sections in the Rational Functions Department of the Function Institute.

1st Degree / 1st Degree Here’s a grapher that shows the behavior of a rational function made up of a first degree polynomial divided by a first degree polynomial. These polynomials are presented in factored form in the following function definition:

Root of numerator: a. Root of denominator: d

1st Degree / 2nd Degree Here’s a grapher that shows the behavior of a rational function made up of a first degree polynomial divided by a second degree polynomial. These polynomials are presented in factored form in the following function definition:

Root of numerator: a Roots of denominator: d, e 1st Degree / 3rd Degree Here’s a grapher that shows the behavior of a rational function made up of a first degree polynomial divided by a third degree polynomial. These polynomials are presented in factored form in the following function definition:

Root of numerator: a Roots of denominator: d, e, f 2nd Degree / 1st Degree Here’s a grapher that shows the behavior of a rational function made up of a second degree polynomial divided by a first degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b Root of denominator: d 2nd Degree / 2nd Degree Here’s a grapher that shows the behavior of a rational function made up of a second degree polynomial divided by a second degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b Roots of denominator: d, e 2nd Degree / 3rd Degree Here’s a grapher that shows the behavior of a rational function made up of a second degree polynomial divided by a third degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b Roots of denominator: d, e, f 3rd Degree / 1st Degree Here’s a grapher that shows the behavior of a rational function made up of a third degree polynomial divided by a first degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b, c Root of denominator: d 3rd Degree / 2nd Degree Here’s a grapher that shows the behavior of a rational function made up of a third degree polynomial divided by a second degree polynomial. These polynomials are presented in factored form in the following function definition:

Roots of numerator: a, b, c Roots of denominator: d, e

Exponential Functions Exponential functions are functions with x as the input variable, and x is in the position of an exponent to a base. That is, these functions, in simple form, look like this: f(x) = 3x The following links explain several aspects of exponential functions.

Definition Transformations of f(x) = 2x Exponential Growth Radioactive Decay Money Matters

Simple Interest Compound Interest Continuous Interest Effective Annual Rate Ordinary Annuity Loans

Transform of f(x) = 2x This Java applet demonstrates several transforms of the function f(x) = 2x. You can assign different values to a, b, h, and k and watch how these changes affect the shape of the graph.

Set a = 1.0, b = 1.0, h = 0.0 and k = 0.0 to see the basic function f(x) = 2x.

Simple Interest Explanation Simple interest is earned on the principal only. Simple interest is not an example of an exponential function. Understanding simple interest, though, will shed light on the understanding of compound interest which is an example of an exponential function.

Formula T Final value of investment P Initial value of investment n

Number of interest periods, usually number of years

r

Percentage rate per interest period, if per year, the annual percentage rate or APR

At start:

T=P After 1 period: T = P + Pr = P(1 + 1r) After 2 periods: T = P + Pr + Pr = P + 2Pr = P(1 + 2r) After 3 periods: T = P + Pr + Pr + Pr = P + 3Pr = P(1 + 3r) After n periods: T = P(1 + nr)

Example calculation If $200 dollars is invested and earns 8.0% simple interest, what is the final value of the investment after 6 years? T = P(1 + nr)

T = 200(1 + (6)(0.080)) T = $296 Calculator for simple interest T = P(1 + nr)

T Final value of investment P Initial value of investment n

Number of interest periods, usually number of years

r

Percentage rate per interest period, if per year, the annual percentage rate (APR)

Compound Interest

Explanation Compound interest is paid both on the original principal and on the accumulated past interest.

Formula S Final value of investment P Initial value of investment i Interest rate per period n Number of interest periods At start:

S=P After 1 period: S = P + Pr = P(1 + i) Amount at the start of this period Amount of interest earned during this period on the starting amount of this period Final value at end of this period After 2 periods:

S = P(1 + i) + P(1 + i)i = P(1 + i)(1 + i) = P(1 + i)2 Amount at the start of this period, i.e., the final value at the end of the last period

Amount of interest earned during this period on the starting amount of this period Final value at end of this period After 3 periods:

S = P(1 + i)2 + P(1 + i)2i = P(1 + i)2(1 + i) = P(1 + i)3 Amount at the start of this period, i.e., the final value at the end of the last period Amount of interest earned during this period on the starting amount of this period Final value at end of this period After n periods:

S = P(1 + i)n Often the interest per period, i, is expressed in terms of the annual percentage rate (APR), r, and the number of interest periods per year, k. Under these conditions the interest per period is equal to the annual percentage rate divided by the number of interest periods per year, as in: i=r/k This would make the above formula for the final value of an investment after n interest periods look like this:

S = P(1 + r/k)n Notice that the output, S, is an exponential function of n. That is, if we consider the final value of the investment as a function of the length of time for the investment, then n, the length of time for the investment, is in the exponent position, and this makes S an exponential function of n.

Example calculation If $4000 is invested at an annual rate of 6.0% compounded monthly, what will be the final value of the investment after 10 years? Since the interest is compounded monthly, there are 12 periods per year, so, k = 12. Since the investment is for 10 years, or 120 months, there are 120 investment periods, so, n = 120. S = P(1 + r/k)n S = 4000(1 + 0.06/12)120 S = 4000(1.005)120 S = 4000(1.819396734)

S = $7277.59 Calculator for compound interest

S = P(1 + r/k)n

S

Final value of investment

P

Initial value of investment

r

Annual percentage rate (APR)

k

Interest periods per year

r/k Percentage rate per interest period n

Number of interest periods

Continuous Interest

Explanation Continuous interest is a form of compound interest. With continuous interest the length of the compounding period is reasoned to be infinitely small. The interest, therefore, is compounded continuously.

Formula S Final value of investment P Initial value of investment r Annual percentage rate (APR) t Number of years Value of investment after t years:

S = Pert Where e is the transcendental number 2.7182818285... Notice that the output, S, is an exponential function of t. That is, if we consider the final value of the investment as a function of the length of time for the investment, then t, the length of time for the investment, is in the exponent position, and this makes S an exponential function of t.

Example calculation If $4000 is invested at an annual rate of 6.0% compounded continuously, what will be the final value of the investment after 10 years? S = Pert

S = 4000e(0.06)(10) S = 4000e0.6 S = 4000(1.822188)

S = $7288.48 Calculator for continuous interest:

S = Pert S Final value of investment P Initial value of investment r APR, Annual percentage rate t Number of years Effective Annual Rate

Explanation The effective annual rate is a value used to compare different interest plans. If two plans were being compared, the interest plan with the higher effective annual rate would be considered the better plan. The interest plan with the higher effective annual rate would be the better earning plan. For every compounding interest plan there is an effective annual rate. This effective annual rate is an imagined rate of simple interest that would yield the same final value as the compounding plan over one year.

Formula S

Final value of investment

P

Initial value of investment

ieff Effective annual rate r

Annual percentage rate (APR)

k

Interest periods per year

After a term of one year the final value, S, of a compounded interest investment with an initial value P compounded k times per year at an annual percentage rate of r is given by:

S = P(1 + r/k)k After a term of one year the final value, S, of a simple interest investment with an initial value P at an annual percentage rate of ieff is given by: S = P(1 + ieff) Setting these two equal we get:

P(1 + ieff) = P(1 + r/k)k Dividing each side by P we get:

(1 + ieff) = (1 + r/k)k And solving for ieff we get: ieff = (1 + r/k)k - 1

Example calculation Which plan is the better investment plan? Plan 1: 8.0% annual percentage rate, compounded monthly Plan 2:

7.9% annual percentage rate, compounded daily ieff for plan 1: ieff = (1 + r/k)k - 1 ieff = (1 + 0.08/12)12 - 1 ieff = (1.006666666)12 - 1 ieff = (1.082999498) - 1 ieff = 0.082999498 ieff = 8.2999% ieff for plan 2: ieff = (1 + r/k)k - 1 ieff = (1 + 0.079/365)365 - 1 ieff = (1.000216438)365 - 1 ieff = (1.082194930) - 1 ieff = 0.082194930 ieff = 8.2194%

Plan 1 is better since 8.2999% > 8.2194%

Calculator for effective annual rate: ieff = (1 + r/k)k - 1

ieff Effective annual rate r

APR, Annual percentage rate

k

Times per year compounded

Ordinary Annuity

Calculator for the future value of an ordinary annuity:

S = R(((1 + i)n - 1) / i) S Final value of annuity R Amount of payment i

Interest rate per period

n Number of payments

Trigonometry Realms sin(x)

Overview •

Quick view of definitions and graphs of trigonometric functions.

Introduction • •

Detailed explanation relating the basic trigonometric functions to the right triangle. Multiple choice problems concerning introductory trigonometry.

Common Angles in a Circle •

Values in degrees and radians for several usual angles from 0 to 360 degrees, (or 0 to 2Pi radians).

Point Definitions for Trig Functions •

Definitions of trig functions based on an (x, y) point on the terminal side of the input angle to the function.

Degrees, Minutes, and Seconds •

How to go back and forth between a decimal number representation of an angle’s measurement and a representation of the measurement in degrees, minutes, and seconds.

The Radian •

Often in mathematics angles are measured in radians rather than in degrees. This section explains the radian and its relationship to the degree.

Solvers •

Right Triangle Solvers

SineThe Graph of the Sine Function

The Right Triangle and the Sine Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The sine of angle A equals the length of the opposite side divided by the length of the hypotenuse. We could write:

sin(A) = opp / hyp So, for example, if the length of the opposite side was 6 and the length of the hypotenuse was 10, then we would write: sin(A) = 6 / 10 sin(A) = 0.6000

CosineThe Graph of the Cosine Function

Above is the graph of the cosine function.

The Right Triangle and the Cosine Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The cosine of angle A equals the length of the adjacent side divided by the length of the hypotenuse. We could write:

cos(A) = adj / hyp So, for example, if the length of the adjacent side was 8 and the length of the hypotenuse was 10, then we would write: cos(A) = 8 / 10 cos(A) = 0.8000

TangentThe Graph of the Tangent Function

The Right Triangle and the Tangent Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The tangent of angle A equals the length of the opposite side divided by the length of the adjacent side. We could write:

tan(A) = opp / adj So, for example, if the length of the opposite side was 6 and the length of the adjacent side was 8, then we would write: tan(A) = 6 / 8 tan(A) = 0.7500

CosecantThe Graph of the Cosecant Function

The Right Triangle and the Cosecant Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The cosecant of angle A equals the length of the hypotenuse divided by the length of the opposite side. We could write:

csc(A) = hyp / opp So, for example, if the length of the hypotenuse was 10 and the length of the opposite side was 6, then we would write: csc(A) = 10 / 6 csc(A) = 1.6667

SecantThe Graph of the Secant Function

The Right Triangle and the Secant Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The secant of angle A equals the length of the hypotenuse divided by the length of the adjacent side. We could write:

sec(A) = hyp / adj So, for example, if the length of the hypotenuse was 10 and the length of the adjacent side was 8, then we would write: sec(A) = 10 / 8 sec(A) = 1.2500

CotangentThe Graph of the Cotangent Function

The Right Triangle and the Cotangent Function

Relative to angle A, this is how the sides of a right triangle would be labeled.

The cotangent of angle A equals the length of the adjacent side divided by the length of the opposite side. We could write:

cot(A) = adj / opp So, for example, if the length of the adjacent side was 8 and the length of the opposite side was 6, then we would write: cot(A) = 8 / 6 cot(A) = 1.3333 Trigonometry and Right Triangles First of all, think of a trigonometry function as you would any general function. That is, a value goes in and a value comes out. If that does not seem quite clear, go see The Definition of a Function and What is f(x)? The names of the three primary trigonometry functions are:

sine cosine tangent These are abbreviated this way:

sine.....sin cosine.....cos tangent.....tan So, instead of writing f(x) , we will write:

sin(x) cos(x) tan(x) Often, in general mathematics notation the parentheses are dropped from the above examples. Therefore, the notation will often look like this:

sin x cos x tan x In Zona Land we will keep the parentheses.

Now, as usual, the input value is x. This input value usually represents an angle. For the sine function, when the input value is 30 degrees, the output value is 0.5. We would write that statement this way: 0.5 = sin(30 °) Below is a listing of several popular input and output values for the three main trigonometry functions. You do not have work at memorizing this table. After you use trigonometry for a while, these values will be remembered quite easily.

0.0000 = sin(0 °)

1.0000 = cos(0 °)

0.0000 = tan(0 °)

0.5000 = sin(30 °) 0.8660 = cos(30 °) 0.5773 = tan(30 °) 0.7071 = sin(45 °) 0.7071 = cos(45 °) 1.000 = tan(45 °) 0.8660 = sin(60 °) 0.5000 = cos(60 °) 1.7320 = tan(60 °) 1.0000 = sin(90 °) 0.0000 = cos(90 °) +infinity = tan(90 °) At this point our central issues will revolve around these questions:

Where do these numbers come from?

What do these numbers mean? Why, for example, is the cosine of 30 degrees equal to 0.8660? The input value for these trigonometric functions is an angle. That angle could be measured in degrees or radians. Here we will consider only input angles measured in degrees from 0 degrees to 90 degrees. This input value appears within the parentheses throughout the above table. The output value for these trigonometric functions is a pure number. That is, it has no unit. This output value appears to the left of the equal sign throughout the above table. There are several ways to understand why a certain input angle produces a certain output value. At first, the most important manner of understanding this is tied to right triangles. All of the trigonometric values for angles between 0 degrees and 90 degrees can be understood by considering this diagram:

We will be concerned angle A. Notice that the sides of the triangle are labeled appropriately “opposite side” and “adjacent side” relative to angle A. The hypotenuse is not considered opposite or adjacent to the angle A. We will also be concerned with length of the three sides. For this discussion we will call the “length of the opposite side” simply the “opposite”. Similarly, the other two lengths will be called “adjacent” and “hypotenuse”. The value for the sine of angle A is defined as the value that you get when you divide the opposite side by the hypotenuse. This can be written: sin(A) = opposite / hypotenuse Or simply: sin(A) = opp / hyp Or, even more simply: sin(A) = o / h Suppose we measure the lengths of the sides of this triangle. Here are some realistic values:

This would mean that: sin(A) = opposite / hypotenuse = 4.00 cm / 7.21 cm = 0.5548 Or simply: sin(A) = 0.5548 Now for the other two trig functions. The value for the cosine of angle A is defined as the value that you get when you divide the adjacent side by the hypotenuse. This can be written: cos(A) = adjacent / hypotenuse Or: cos(A) = adj / hyp Or: cos(A) = a / h Using the above measured triangle, this would mean that: cos(A) = adjacent / hypotenuse = 6.00 cm / 7.21 cm = 0.8322 Or simply: cos(A) = 0.8322 The value for the tangent of angle A is defined as the value that you get when you divide the opposite side by the adjacent side. This can be written: tan(A) = opposite / adjacent Or: tan(A) = opp / adj

Or: tan(A) = o / a Using the above measured triangle, this would mean that: tan(A) = opposite / adjacent = 4.00 cm / 6.00 cm = 0.6667 Or simply: tan(A) = 0.6667 The angle A in the above triangle is actually very close to 33.7 degrees. So, we would say:

0.5548 = sin(33.7°) 0.8322 = cos(33.7°) 0.6667 = tan(33.7°) So, suppose that you wanted to know the trigonometry values for 47.5 degrees? You could carefully draw a right triangle using a ruler and protractor that had an angle equal to 47.5 degrees in the position of angle A. Then, you could carefully measure the sides. Lastly you could divide the appropriate sides to find the values for the three trigonometric functions. You would find that:

0.7373 = sin(47.5°) 0.6755 = cos(47.5°) 1.0913 = tan(47.5°) Someone has already done this, in a way, for all the possible angles. All the input angles and output values are listed in tables called trig tables. They look like this:

Angle

sin

cos

tan

0.0

0.0000

1.0000 0.0000

0.5

0.0087

0.9999 0.0087

1.0

0.0174

0.9998 0.0174

And so on...

These let you look up the trigonometric values for any angle. Calculators and computers, of course, will let you do the same.

Here is a demonstration that shows you these trig calculations for several angles. Use the slider to adjust the size of the angle. Notice how the values are calculated for each trig function depending upon the lengths of the sides of the triangle. Below is again the triangle from the above diagrams, except now the other acute angle, B, is marked. Also marked are the sides that are adjacent and opposite to angle B.

Here are the three trig functions for angle B:

sin(B) = opposite / hypotenuse = 6.00 cm / 7.21 cm = 0.8322 cos(B) = adjacent / hypotenuse = 4.00 cm / 7.21 cm = 0.5548 tan(B) = opposite / adjacent = 6.00 cm / 4.00 cm = 1.5000 If you look carefully you will notice that the sine of angle B is the same value we calculated above for the cosine of angle A. You should also notice that the cosine of angle B is equal to previous calculation for the sine of angle A. This is because the opposite side for angle B is the adjacent side for angle A, and because the adjacent side for angle B is the opposite side for angle A. This is demonstrated in the following diagram:

We could summarize this relationship this way:

sin(A) = A’s opposite / hypotenuse = 4.00 cm / 7.21 cm = 0.5548 cos(B) = B’s adjacent / hypotenuse = 4.00 cm / 7.21 cm = 0.5548 cos(A) = A’s adjacent / hypotenuse = 6.00 cm / 7.21 cm = 0.8322 sin(B) = B’s opposite / hypotenuse = 6.00 cm / 7.21 cm = 0.8322 Now, angle A and B form a pair of complementary angles. That is, their measurements add up to 90 degrees. This is because the measurement of the interior angles for any triangle must sum to 180 degrees, and in this triangle 90 of those degrees are taken up by the right angle, so that leaves 90 degrees remaining from the total of 180 to be split up between angle A and B. So, here we notice that the sine of an angle is equal to the cosine of its complement, and that the cosine of an angle is equal to the sine of its complement. Also, we will take note of the relationship between the tangents of the complementary angles A and B. The tangent of angle A is equal to the reciprocal, or inverse, of the tangent of angle B, and, likewise, the tangent of angle B is equal to the reciprocal of the tangent of its complement, angle A. This is summarized in the following table:

tan(A) = A’s opposite / A’s adjacent = 4.00 cm / 6.00 cm = 0.6667 tan(B) = B’s opposite / B’s adjacent = 6.00 cm / 4.00 cm = 1.5000 Here is an easy way to remember these relationships for trig functions and the right triangle. Just write down this mnemonic: SOH - CAH - TOA It is pronounced “so - ka - toe - ah”. The SOH stands for “Sine of an angle is Opposite over Hypotenuse.”

The CAH stands for “Cosine of an angle is Adjacent over Hypotenuse.” The TOA stands for “Tangent of an angle is Opposite over Adjacent.” What is f(x) ?

Common Angles Around a Circle The circle below can be thought of as being divided into angles that are integer multiples of 30, 45, 60, and 90 degree angles. Click one of the points on the circle to see the angle and its measurement in both degrees and radians. See notes below. The angles marked on this circle represent common angles that are often used in introductory geometry and trigonometry problems. Try figuring out the angle measurement before you click on the point. Then use this applet to check your value. All of the angles will be shown as arcs when they are drawn. You should notice that all of these angles are in standard position. This Java applet will not print the value for pi as a Greek letter. Instead, it is printed as ‘Pi’. This, of course, approximately equals the value 3.14. Trig Function Point Definitions Quick Instructions: • •

The above applet starts with the sine function active. Any of the six trigonometric functions can be activated by choosing the appropriate radio button at the top of the applet. The large square graph on the left is the (x, y) coordinate plane. It extends from 10.0 to +10.0 along the x-axis and the y-axis.

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The upper right rectangular graph shows the graph of the currently active trig function. For example, when the sine function is active, this graph shows the sin(a) vs. a, where a represents the angle. Horizontally, this graph has a domain from 0 radians to 2pi radians with markers every pi/2 radians. Vertically, the range of this graph runs from -2.0 to +2.0 for the sine and cosine functions, and it runs from -5.0 to +5.0 for the tangent, cotangent, secant and cosecant functions. Vertical marking occur every unit distance. The lower right rectangular area shows the calculation for the currently active trig function at the current (x, y) point on the left graph. When the sine function radio button is selected, click somewhere on the left (x, y) graph. Notice: o The (x, y) coordinates are presented above the selected point. o The angle whose terminal side goes through this (x, y) point is drawn. Its value is presented in the upper right corner of this (x, y) graph. o The relevant quantities necessary for the sine calculation are drawn on the graph, (the y distance and the radius in this case). The values for these quantities are presented near their graphic representations. o The upper right hand graph shows the current function, i. e., the sine function. The current input angle and the current output value for this function are presented. o In the lower right rectangle of the applet the value for the current trig function at the current angle is calculated using the the relevant quantities from the (x, y) graph. All values are rounded to two decimal places. Certainly more precise values for the trig functions are available elsewhere. This applet, though, is not meant to be a calculator. It is meant to demonstrate the interrelationships of several trigonometric concepts. Try different (x, y) positions and different trig functions.

Further Discussion: This material explains the definitions of the six trigonometric functions in terms of an (x, y) point located on the terminal side of the input angle. You should be familiar with: • The (x, y) coordinate plane. • How to find the distance from the origin to an (x, y) point. • The graphs of the trig functions. • Angles in standard position. • Radian measure for angles. The six trigonometric functions, (sine, cosine, tangent, cotangent, secant and cosecant), are usually thought to accept an angle as input and output a pure number. For the purposes of the definitions this angle is to be placed in standard position. We will be concerned with any (x, y) point located on the terminal side of this angle. These definitions are based on such an (x, y) point. These definitions also use the distance from the origin to the (x, y) point. This distance will be referred to as r and can be calculated like this:

The six definitions are: sin(angle) = y/r cos(angle) = x/r tan(angle) = y/x (x not equal to zero) csc(angle) = r/y (y not equal to zero) sec(angle) = r/x (x not equal to zero) cot(angle) = x/y (y not equal to zero) So, for example, the point (5, 7) is on the terminal side of an angle in standard position which has a measure of about 0.95 radians (about 54 degrees):

The distance from the origin to the point (5, 7) can be calculated this way (approximate result given):

Therefore, the six trigonometric function values for this angle can be calculated as follows (approximate results given):

sin(0.95) = y/r = 7/8.6 = 0.81 cos(0.95) = x/r = 5/8.6 = 0.58 tan(0.95) = y/x = 7/5 = 1.4

csc(0.95) = r/y = 8.6/7 = 1.2 sec(0.95) = r/x = 8.6/5 = 1.7 cot(0.95) = x/y = 5/7 = 0.71 In all of the above calculations approximate results were given. Of course, if you are doing more careful and important work you would use measurements and calculations of higher significance. The definitions given here for the six trigonometric functions are more powerful than the right triangle definitions given in the introduction to trigonometry section. The right triangle definitions are only good for angles up to pi/2 radians (90 degrees). These trig definitions based upon an (x, y) point on the terminal side of an angle are good for angles of any measurement, positive or negative.

Degrees, Minutes, Seconds

There are several ways to measure the size of an angle. One way is to use units of degrees. (Radian measure is another way.)

In a complete circle there are three hundred and sixty degrees. An angle could have a measurement of 35.75 degrees. That is, the size of the angle in this case would be thirty-five full degrees plus seventy-five hundredths, or three fourths, of an additional degree. Notice that here we are expressing the measurement as a decimal number. Using decimal numbers like this one can express angles to any precision - to hundredths of a degree, to thousandths of a degree, and so on. There is another way to state the size of an angle, one that subdivides a degree using a system different than the decimal number example given above. The degree is divided into sixty parts called minutes. These minutes are further divided into sixty parts called seconds. The words minute and second used in this context have no immediate connection to how those words are usually used as amounts of time.

In a full circle there are 360 degrees. Each degree is split up into 60 parts, each part being 1/60 of a degree. These parts are called minutes. Each minute is split up into 60 parts, each part being 1/60 of a minute. These parts are called seconds.

The size of an angle could be stated this way: 40 degrees, 20 minutes, 50 seconds. There are symbols that are used when stating angles using degrees, minutes, and seconds. Those symbols are show in the following table.

Symbol for degree: Symbol for minute: Symbol for second: So, the angle of 40 degrees, 20 minutes, 50 seconds is usually written this way:

How could you state the above as an angle using common decimal notation? The angle would be this many degrees, (* means times.):

40 + (20 * 1/60) + (50 * 1/60 * 1/60) That is, we have 40 full degrees, 20 minutes - each 1/60 of a degree, and 50 seconds each 1/60 of 1/60 of a degree. Work that out and you will get a decimal number of degrees. It’s 40.34722... Going the other way is a bit more difficult. Suppose we start with 40.3472 degrees. Can we express that in units of degrees, minutes, and seconds? Well, first of all there are definitely 40 degrees full degrees. That leaves 0.3472 degrees. So, how many minutes is 0.3472 degrees? Well, how many times can 1/60 go into 0.3472? Here’s the same question: What is 60 times 0.3472? It’s 20.832. So, there are 20 complete minutes with 0.832 of a minute remaining. How many seconds are in the last 0.832 minutes. Well, how many times can 1/60 go into 0.832, or what is 60 times 0.832? It’s 49.92, or almost 50 seconds. So, we’ve figured that 40.3472 degrees is almost exactly equal to 40 degrees, 20 minutes, 50 seconds. (The only reason we fell a bit short of 50 seconds is that we really used a slightly smaller angle in this second half of the calculation explanation. In the original angle, 40.34722... degrees, the decimal repeats the last digit of 2 infinitely, so, the original angle is a bit bigger than 40.3472.)

The Radian The word radian describes a certain size of an angle. In the Java animation below the blue pie slice shape demonstrates an angle of one radian. Notice that for an angle of one radian the arc length along the edge of the circle is equal in length to the radius. Read the details below... Usually, a person first learns how to measure the size of an angle using degrees. There are, of course, 360 degrees all the way around a circle. The degree, however, turns out to be not a very mathematical way of measuring angles. Perhaps the degree was originally created as approximately the angle which the Earth goes through per day as it orbits the Sun, since there are about 360 days in the year. The radian is used much more than the degree in higher mathematics for measuring angles. A radian is defined this way: • •

If you have a circle with an angle whose vertex is at the center of that circle... And if that angle is of such a size that the amount of the circumference of the circle which that angle intercepts has an arc length equal to the length of the radius...

• Then that angle has a measurement of one radian. Here’s another way to say it: A central angle to a circle has a size of one radian if it subtends an arc length on the circle equal in length to the radius. In other words, if you could pick up the radius of a certain circle like it were a plastic rod and bend it around the circumference of that circle, then that bent radius length would touch the sides of a central angle which had a measurement of one radian. Look at the following picture and then go back and look at the animation.

One thing to understand about a radian is that it is bigger than a degree. In fact: 1 radian = 57.2957 degrees 1 degree = 0.0174532 radians The above values are to six significant figures, truncated, not rounded. If you want to know the exact size of the radian in terms of degrees, take 360 and divide it by 2 times pi. That number is how many degrees there are in a radian. What is the reasoning behind these values? It works this way: How many times can you trace an arc length equal to the size of the radius as you move around the circumference of a circle? Well, here’s the formula for the circumference of a circle:

C is the circumference, and r is the radius. Looks like there are two pi radii around the circumference of a circle, or about 2 times 3.14, i.e., about 6.28, radii around the circumference of a circle. Each one of these radius lengths would designate one radian, so there are about 6.28 radians in a full circle. The following diagram shows this:

Therefore, there are 2 pi radians in a full circle. We also know that there are 360 degrees in a circle. So, there are 360 degrees per 2 pi radians. Dividing 360 by 2 pi give us the value of about 57.2957 degrees per radian. A radian is equal to 57.2957 degrees. Also, by dividing 2 pi radians by 360 degrees we get about 0.0174532 radians per degree. A degree is equal to 0.0174532 radians. In mathematics if you state the size of an angle as a pure number, without the degree ‘unit’ marker after it, then the angle is taken to be in radians. So, if the angle in question is named A, and if someone were to write down:

A=4 Then that would mean that angle A has a measurement of 4 radians and not 4 degrees. Trigonometric Functions - Right Triangle Solvers Here is a group of solvers that work with the following right triangles.

In each of the above diagrams the acute angles of the right triangle are named angle D and angle E. The right angle is not named. The two sides, or legs, of the triangle have lengths named d and e. The length of the hypotenuse is named f. In each case you will be starting with known values for two parts of the right triangle. For example, in the first triangle you are starting with known values for angle D and side length d. You are to create these initial known values. In each case you are to find the other parts of the right triangle. For example, in the first triangle you are to find values for angle E and the lengths of sides d and e and the length of the hypotenuse f.

Intersections

Intersection of Two Lines Intersection of a Line and a Parabola Intersection of Two Parabolas

Intersection of Two Lines

Here we will cover a method for finding the point of intersection for two linear functions. That is, we will find the (x, y) coordinate pair for the point were two lines cross. Our example will use these two functions: f(x) = 2x + 3 g(x) = -0.5x + 7 We will call the first one Line 1, and the second Line 2. Since we will be graphing these functions on the x, y coordinate axes, we can express the lines this way: y = 2x + 3 y = -0.5x + 7 Those two lines look this way:

Now, where the two lines cross is called their point of intersection. Certainly this point has (x, y) coordinates. It is the same point for Line 1 and for Line 2. So, at the point of intersection the (x, y) coordinates for Line 1 equal the (x, y) coordinates for Line 2.

Since at the point of intersection the two y-coordinates are equal, (we’ll get to the xcoordinates in a moment), let’s set the y-coordinate from Line 1 equal to the y-coordinate from Line 2. The y-coordinate for Line 1 is calculated this way: y = 2x + 3 The y-coordinate for Line 2 is calculated this way: y = -0.5x + 7 Setting the two y-coordinates equal looks like this:

2x + 3 = -0.5x + 7 Now, we do some algebra to find the x-coordinate at the point of intersection:

2x + 3 = -0.5x + 7

We start here.

2.5x + 3 = 7

Add 0.5x to each side.

2.5x = 4

Subtract 3 from each side.

x = 4/2.5

Divide each side by 2.5.

x = 1.6

Divide 4 by 2.5.

So, we have the x-coordinate for the point of intersection. It’s x = 1.6. Now, let’s find the y-coordinate. The y-coordinate can be found by placing the x-coordinate, 1.6, into either of the equations for the lines and solving for y. We will first use the equation for Line 1:

y = 2x + 3 y =2(1.6) + 3 y = 3.2 + 3 y = 6.2 Therefore, the y-coordinate is 6.2. To make sure our calculations are correct, and also to demonstrate a point, we should get the same y-coordinate if we use the equation for Line 2. Let’s try that: y = -0.5x + 7 y = -0.5(1.6) + 7 y = -0.8 + 7 y = 6.2 Well, looks like everything has worked out. The point of intersection for these two lines is (1.6, 6.2). If you look back at the graph, this certainly makes sense:

Here’s the summary of our methods: 1. Get the two equations for the lines into slope-intercept form. That is, have them in this form: y = mx + b. 2. Set the two equations for y equal to each other. 3. Solve for x. This will be the x-coordinate for the point of intersection. 4. Use this x-coordinate and plug it into either of the original equations for the lines and solve for y. This will be the y-coordinate of the point of intersection. 5. As a check for your work plug the x-coordinate into the other equation and you should get the same y-coordinate. 6. You now have the x-coordinate and y-coordinate for the point of intersection. Actually, there is nothing special about the functions being linear functions. This method could be used to find the point or points of intersection between many other types of functions. One would express the functions in ‘y =’ form, set the right side of these forms equal to each other, solve for x, (or x’s), and use this x, (or x’s), to find the corresponding y, (or y’s).

Intersection of Line and Parabola

Here we will cover a method for finding the point or points of intersection for a linear function and a quadratic function. Quadratic functions graph as parabolas. So, we will find the (x, y) coordinate pairs where a line crosses a parabola. First, understand that a line and a parabola may intersect at two points, as in this picture:

Or, they may intersect at only one point, as in this picture:

And, they may not meet at any points, as shown here:

For this example we will use these two functions: f(x) = 1.5x + 5 g(x) = 2x2 + 12x + 13 The first one, f(x), is a line, and the second, g(x), is a parabola. Since we will be graphing these functions on the x, y coordinate axes, we can express them this way: y = 1.5x + 5 y = 2x2 + 12x + 13 When graphed, these two functions look like this:

Now, the places where the two functions cross are called their points of intersection. At these points of intersection the x-coordinate for the line equals the x-coordinate for the parabola, and the y-coordinate for the line equals the y-coordinate for the parabola.

Since at the points of intersection the y-coordinates are equal, (we’ll get to the xcoordinates in a moment), let’s set the y-coordinate from line equal to the y-coordinate from parabola. The y-coordinate for the line is calculated this way: y = 1.5x + 5 The y-coordinate for the parabola is calculated this way: y = 2x2 + 12x + 13 Setting the two y-coordinates equal looks like this:

1.5x + 5 = 2x2 + 12x + 13 When we solve the above equation, we find the x-coordinates for the points of intersection. Here’s the algebra:

We start here. Subtract 1.5x from each side. Subtract 5 from each side. Use the quadratic formula to solve for x. Here are are two solutions for x. Now, we will plug each value for the x-coordinate into either of the intersecting functions to get its corresponding y-coordinate. Let’s first work with x = -0.92. We will plug this into the equation of the line to get the y-coordinate. Here is that work: y = 1.5x + 5 y = 1.5(-0.92) + 5 y = -1.38 + 5 y = 3.62 Notice that if we would have chosen to have worked with the quadratic function, we would have gotten the same y-coordinate. (We are off just a bit due to rounding and truncating decimals along the way, both above in the evaluation of the quadratic formula and below where we keep intermediate results to two decimals.): y = 2x2 + 12x + 13 y = 2(-0.92)2 + 12(-0.92) + 13 y = 1.69 - 11.04 + 13 y = 3.65 So, to a fair degree of accuracy, the two functions intersect at (-0.92, 3.62). Let’s find the other point of intersection. We know that the x-coordinate is -4.33, and using the equation for the line we can find the y-coordinate: y = 1.5x + 5 y = 1.5(-4.33) + 5 y = -6.49 + 5 y = -1.49

Therefore, the two functions also intersect at (-4.33, -1.49). Here’s how those spots look on the graph:

You can use the above procedure can be used to find the intersection of any line with any parabola. Of course, the line and parabola will not always intersect at two points. Sometimes they will only intersect at one point, and quite often they will not intersect at all. These conditions will show up when you solve the quadratic equation after you set the two separate functions equal to each other and collect like terms. When solving that quadratic equation, if the discriminant equals zero, then you will have only one solution for x, which boils down to only one point of intersection. If the discriminant is negative, then there are no solutions for x, which means the two parabolas do not intersect. Want to try this yourself? Below is a calculator which you can use to check your work. First, make up two functions, one for a line and one for a parabola. You could use the function graphers in the Function Institute to help you find the values for the first degree polynomial, (linear function for the line), and the second degree polynomial, (quadratic function for the parabola), or you could just make them up. Perhaps you would choose: y = 1.5x + 5 (for the line) y = 2x2 - 12x + 13 (for the parabola) If you wanted to see these functions using EZ Graph you would type each into one of the ‘y = ‘ boxes at the bottom of the grapher like this: 1.5*x + 5 2*x^2 - 12*x + 13 Intersection of Two Parabolas

We will cover a method for finding the point or points of intersection for two quadratic functions. Quadratic functions graph as parabolas. So, we will find the (x, y) coordinate pairs where the two parabolas intersect. First, understand that two parabolas may intersect at two points, as in these pictures:

Or, the parabolas may intersect at only one point, as in these pictures:

And, they may not meet at any points, as shown here:

Our example will use these two quadratic functions: f(x) = 1.5x2 - 9x + 11.5 g(x) = -0.2x2 - 0.4x + 2.8 Of course, quadratic functions, or second degree polynomial functions, graph as parabolas. Since we will be graphing these functions on the x, y coordinate axes, we can express the parabolas this way: y = 1.5x2 - 9x + 11.5 y = -0.2x2 - 0.4x + 2.8 Those two parabolas look this way:

Now, where the two parabolas cross is called their points of intersection. Certainly these points have (x, y) coordinates, and at the points of intersection both parabolas share the same (x, y) coordinates. So, at the points of intersection the (x, y) coordinates for f(x) equal the (x, y) coordinates for g(x).

Since at the points of intersection the y-coordinates are equal, (we’ll be working with the x-coordinates later), let’s set the y-coordinate from from one parabola equal to the ycoordinate from the other parabola. The y-coordinate for f(x) is calculated this way: y = 1.5x2 - 9x + 11.5 The y-coordinate for g(x) is calculated this way: y = -0.2x2 - 0.4x + 2.8 Setting the two y-coordinates equal to each other creates this equation:

1.5x2 - 9x + 11.5 = -0.2x2 - 0.4x + 2.8 Now, we do some algebra to find the x-coordinates at the points of intersection:

This is where we start. Subtract 1.5x2 from each side. Add 9x to each side. Subtract 11.5 from each side. This leaves us with a quadratic equation. Use the quadratic formula to solve for x.

These are the xcoordinates for the two points of intersection. (The values her are truncated to two decimal places and not rounded.) So, we have the x-coordinates for the points of intersection. They are x = 1.39 and x = 3.66. Now, let’s find the y-coordinates. Each y-coordinate can be found by placing its corresponding x-coordinate into either of the equations for the parabolas and solving for y. We will first use the equation from the first parabola. Here we plug x = 1.39 into it: y = 1.5x2 - 9x + 11.5 y = 1.5(1.39)2 - 9(1.39) + 11.5 y = 2.89 - 12.51 + 11.5 y = 1.88 So, one point of intersection is very close to (1.39, 1.88). Here, we say very close to since the values have been calculated using only two decimal places. Now, plugging in the other x-coordinate into the equation for the first parabola, we can get the other y-coordinate for the second point of intersection: y = 1.5x2 - 9x + 11.5 y = 1.5(3.66)2 - 9(3.66) + 11.5 y = 20.09 - 32.94 + 11.5 y = -1.35 The other point of intersection is very near (3.66, -1.35). Here are these points of intersection shown on the graph of the two parabolas:

The above procedure can be used to find the intersection of any two parabolas. Of course, the parabolas will not always intersect at two points. Sometimes they will only intersect

at one point, and quite often they will not intersect at all. These conditions will show up when you solve the quadratic equation after you set the two separate functions equal to each other and collect like terms. When solving that quadratic equation, if the discriminant equals zero, then you will have only one solution for x, which boils down to only one point of intersection. If the discriminant is negative, then there are no solutions for x, which means the two parabolas do not intersect. Ready to try it yourself? Below is a calculator which you can use to check your work. First, make up two second degree polynomial functions, (quadratic functions), in this form: y = ax2 + bx + c You could use the function grapher in the Function Institute to help you find the values for a, b, and c, or you could just make them up. Perhaps for one of your parabolas you would choose: y = 1.5x2 - 9x + 11.5 If you wanted to see this function using EZ Graph you would type this into one of the ‘y = ‘ boxes at the bottom: 1.5*x^2 - 9*x + 11.5

The Geometry Section

Common Shapes Areas and Volumes Coordinate Geometry Points, Lines, Planes

Common Shapes

Two Dimensional Circle Rhombus Trapezoid The Circle

The circle is a set of points equally distant from one central point. This central point is called the center of the circle. Here the center is shown with a small cross. The distance from the center to the curve of the circle is called the radius. Here the radius is shown as a line from the center to the curve of the circle.

The distance across the circle is called the diameter. The diameter always passes through the center of the circle. The diameter is equal in length to twice the radius.

The distance all the way around the circle is called the circumference. The circumference is usually symbolized with a ‘C’ as shown here. The length of the circumference is equal to 2 times pi times the radius. Usually, as here, the radius is symbolized with an ‘r’. The Rhombus

The rhombus is a quadrilateral with all sides equal in length. Notice that it is a parallelogram, since opposite sides are parallel.

Since the rhombus is a parallelogram, you find its area using the same method that you would use for a parallelogram. The length of the base of the rhombus is the length of one of its sides, here shown with ‘b’. The height is the perpendicular distance between opposite sides, here shown with an ‘h’. The area of the rhombus is equal to its base times its height, or: area = bh

The diagonals of a rhombus are perpendicular.

A square is a rhombus since it is a quadrilateral with all sides equal.

The Trapezoid

The trapezoid is a quadrilateral with one pair of parallel sides.

The parallel sides of a trapezoid are called the bases, here symbolized by b1 and b2. The height of the trapezoid is the perpendicular distance between the bases, here symbolized by h. The area of the trapezoid is equal to the average of the bases times the height. So, you add the two bases, divide by 2, and then multiply by the height. This would be: area = ((b1 + b2) / 2)h

If the two sides which are not parallel have equal lengths, then the trapezoid is called an isosceles trapezoid. The base angles in an isosceles trapezoid are equal in measurement.

Another example of a trapezoid.

Yet another example of a trapezoid.

Total Surface Areas and Volumes of Simple Solids Box Cylinder Box with Cylindrical Hole Total Surface Area and Volume of a Box

The total surface area is made up of three pairs of sides for a total of six sides. We must find Areahw, which is the area of the side that is h by w.

Areahw = (h)(w) We must find Areahl, which is the area of the side that is h by l. Areahl = (h)(l) We must find Areawl, which is the area of the side that is w by l. Areawl = (w)(l) Each of the above areas are present on two sides of the box. So, the total surface area is sum of twice each of these areas, that is: Total Surface Area = 2(Areahw) + 2(Areahl) + 2(Areawl) The volume of the box is simply the product of h, w, and l. Volume = (h)(w)(l) Example: Let h = 3 cm, w = 5 cm, l = 8 cm Areahw = (h)(w) = (3 cm)(5 cm) = 15 cm2 Areahl = (h)(l) = (3 cm)(8 cm) = 24 cm2 Areawl = (w)(l) = (5 cm)(8 cm) = 40 cm2 Total Surface Area = 2(Areahw) + 2(Areahl) + 2(Areawl) Total Surface Area = 2(15 cm2) + 2(24 cm2) + 2(40 cm2) Total Surface Area = 30 cm2 + 48 cm2 + 80 cm2 Total Surface Area = 158 cm2 Volume = (h)(w)(l) Volume = (3 cm)(5 cm)(8 cm) Volume = 120 cm3 Total Surface Area and Volume of a Cylinder

The total surface area is made up of a circular top and bottom, which are each the same size, and the side surface, which is actually a rectangle. We must find the area of the top, or Areatop, which is the area of a circle with a radius of r. Areatop = (pi)(r2) This area is the same as the area of the bottom, so: Areabottom = Areatop The area of the side is actually a rectangle. Imagine the cylinder to be a simple tin can. In your mind, cut off the top and the bottom and discard them. What is left is an open ended tube. Again in your mind, use some tin snips to cut along a straight line from top to bottom and uncurl and flatten out the tube. You would be looking at a rectangle that is as tall as the cylinder and as long as the circumference of its top (or bottom). Therefore, the area of the side is: Areaside = (h)(Circumferencetop) Now, since the circumference of the top is: Circumferencetop = 2(pi)® The area of the side is: Areaside = (h)(Circumferencetop) Areaside = (h)(2(pi)®) Areaside = 2(h)(pi)® So, the total area of the cylinder is equal to the top area plus the bottom area plus the side area:

Total Surface Area = Areatop + Areabottom + Areaside The volume of the cylinder, since it is a right solid, is the product of the top area times the height.

Volume = (Areatop)(h)

Example: Let h = 5 cm, r = 2 cm Areatop = (pi)(r2) Areatop = (3.14)(2 cm)2 = (3.14)4 cm2 = 12.56 cm2

Areabottom = Areatop = 12.56 cm2 Areaside = (h)(Circumferencetop) Areaside = (h)(2(pi)®) Areaside = 2(h)(pi)® Areaside = 2(5 cm)(3.14)(2 cm) Areaside = 62.8 cm2 Total Surface Area = Areatop = + Areabottom + Areaside Total Surface Area = 12.56 cm2 + 12.56 cm2 + 62.8 cm2 Total Surface Area = 87.92 cm2 Volume = (Areatop)(h) Volume = (12.56 cm2)(5 cm) Volume = 62.8 cm3 Total Surface Area and Volume of a Box with a Cylindrical Hole

Be sure you understand how to find the total surface area and volume of a box and of a cylinder before going ahead here. The total surface area of this new shape is made up several areas, some like parts of the box, some like parts of the cylinder, and some like differences between parts of the box and cylinder. Like the box, this shape has a pair of sides with areas which we will again call Areahw. Like the box, this shape also has another pair of sides with areas which we will again call Areahl. The top and bottom of this shape form another pair of equal areas. These areas are both equal to the area of the top of the box after a circular hole has been cut out of it. This hole

has an area equal to the area of the circular top of the cylinder. So, the area of the top of this shape is calculated this way:

Areatop = AreaboxTop - AreacylinderTop The area of the top of the box is Arealw, so, restating the above:

Areatop = Arealw - AreacylinderTop Again, we have two areas, top and bottom, that are equal to the immediate above value. In this shape the area of the side of the hole is like the area of the side of the cylinder, so: AreaholeSide = AreacylinderSide So, the total area of this shape is equal to: Areatotal = 2(Areahw) + 2(Areahl) + 2(Areatop) • AreaholeSide The volume of this shape is the volume of the box, as calculated before, minus the volume of the cylindrical hole, also as calculated before.

Volume = Volumebox - Volumecylinder Coordinate Geometry

VRML (x, y, z) Axes and Planes

VRML (x, y, z) Axes and Planes Worsslds These worlds illustrate the (x, y, z) coordinate axes and the three planes associated with these axes. The axes and planes all extend to infinity. However, in these demonstrations they are drawn to a finite size. Use the Back button of your browser to return from them.

Conic Sections A conic section is a section of a cone. The popular ellipse, parabola, and hyperbola, along with a few other mathematical shapes, can each be seen to be a section of a cone.

In this context, the cone is thought to be a hollow cone, quite a bit like the ‘sugar cones’ in an ice cream shop. So, these mathematical cones look like this one:

And actually when thinking about conic sections, we envision two such cones lined up vertically tip to tip. So, conic section cones in mathematics look like this:

We must think of these cones as going on forever without a top or bottom limit.

(Such a pair of cones is formally called a circular conical surface, and a discussion of conic sections is truly centered around such a surface.) A line which we imagine running through the center of the cones in a direction perpendicular to their bases is called the axis. This is shown below:

Now, about the section part in the term conic section. This section is a very thin slice of the cones. In fact, it is an infinitely thin slice. One thinks of the cones as being sliced by a plane. So, we speak about the intersection of the cones with a plane. The shape of this intersection is the shape of the conic section. Below is one example of how we could imagine a cone being intersected by a plane. (Of course, both the cone and the plane actually extend to infinity.)

The shape of the intersection, or cut, that the plane makes with the cone is the shape of the conic section. For example, in the above picture the intersection is a hyperbola. The way in which the plane cuts through the cone determines the particular conic section. It determines if the conic section is a parabola or ellipse, and so on. Several ways in which a plane can intersect the cone are illustrated in the following VRML animation. (If you have any problem viewing VRML in your browser, you probably need a VRML plug-in, something which is well worth getting. You can get help with that here.) This and the other VRML animations listed below show cones with bottoms rather than showing hollow cones. As it turns out, this makes it much easier to see cones in VRML.

Conic Sections - The Ellipse On an (x, y) graph an ellipse can have this shape:

If a plane intersects the cone all of the way around but is not perpendicular to the axis, the conic section is an ellipse. This is shown below:

Conic Sections - The Parabola On an (x, y) graph a parabola can have this shape:

If a plane intersects the cone when it is slanted the same as the side of the cone, (formally, when it is parallel to the slant height), the conic section is a parabola. This is shown below:

Conic Sections - The Hyperbola On an (x, y) graph a hyperbola can have this shape:

If a plane intersects the cone when it is parallel to the axis, the conic section is a hyperbola. This is shown below:

Conic Sections - The Circle

On an (x, y) graph a circle can have this shape:

If a plane intersects the cone when it is perpendicular to the axis, the conic section is a circle. Actually, a circle is a special case of an ellipse. This is shown below:

Conic Sections - The Line

On an (x, y) graph a line can have this shape:

If a plane intersects the cone when it is slanted the same as the side of the cone, (formally, when it is parallel to the slant height), and when it passes through the tip of the cone, (formally when it passes through the vertex), the conic section is a line. This is shown below:

Conic Sections - The Point

On an (x, y) graph a point has this shape:

If a plane intersects the cone when it is perpendicular to the axis and when it passes through the tip of the cone, the conic section is a point. Actually, in this discussion a point is a special case of a circle. This is shown below:

Simple Data Grapher Here is a very simple data grapher. Basically, you enter your legends, maximums, minimums, and data, and then you press the ‘Make graph’ button. A new browser window will open up with your graph displayed in it. Simply print the page using the services of the new browser window, and you will have a hard copy of your graph. Close the new browser window when you are done. This page remains displayed in this original browser window. Suggestions and help: • • •

Data points are plotted with this character: ‘*’. Use font size 1. Graphs made with this small font seem to print best on most equipment. The independent axis is the horizontal axis, usually called the x-axis in algebra. (Beware: In physics x usually means position and is often graphed vertically.). The dependent axis is the vertical axis, usually called the y-axis in algebra.

•

•

•

The syntax for entering data points works like this: If you want to enter point (4.2, 8.5), then on one line in the text box used for data entry write ‘4.2, 8.5;’ without the quotes. That is, type in four point two comma eight point five semicolon. The comma between coordinates and the semicolon at the end are important. Put one data point on one line in the text box.

• •

You can graph up to 100 points. Round your minimums and maximums so they easily extend over the range of your data. The difference between each minimum and its corresponding maximum should be evenly divisible by 5 for the graph tic mark values on the axes to look good. • This graph maker works with a fixed width text font, and therefore the placement of the points is not perfect; it is rounded to the nearest text row and column. The graphs are hardly useless, however. You will get a good idea of how your data looks, and the graphs are meant to be easy to print on almost all equipment. • If you want to see how the whole thing works, just click the ‘Make graph’ button. The grapher defaults to a simple position vs. time, or x vs. t, graph. 45-45-90 and 30-60-90 Triangles This page summarizes two types of right triangles which often appear in the study of mathematics and physics. One of these right triangles is named a 45-45-90 triangle, where the angles in the triangle are 45 degrees, 45 degrees, and 90 degrees. This is an isosceles right triangle. The other triangle is named a 30-60-90 triangle, where the angles in the triangle are 30 degrees, 60 degrees, and 90 degrees. Common examples for the lengths of the sides are shown for each below.

The 45-45-90 Triangle

Here we check the above values using the Pythagorean theorem. The length of the hypotenuse should be equal to the square root of the sum of the squares of the legs of the triangle.

Listed below are the values shown in the diagram as well as another common set of values for this triangle. Be sure to notice that the two legs have the same length, so, the leg length is listed only once.

Hypotenuse Leg Length Length 1

0.7071

1.4142

1

The 30-60-90 Triangle

Here we check the above values using the Pythagorean theorem. The length of the hypotenuse should be equal to the square root of the sum of the squares of the legs of the triangle.

Listed below are the values shown in the diagram as well as another common set of values for this triangle.

Hypotenuse Leg Length Leg Length Length Opposite 30 Deg Opposite 60 Deg 1

0.5000

0.8660

2

1

1.7320

1.1547

0.5773

1

Interval Notation Interval notation is a method of writing down a set of numbers. Usually, this is used to describe a certain span or group of spans of numbers along a axis, such as an x-axis. However, this notation can be used to describe any group of numbers. For example, consider the set of numbers that are all greater than 5. If we were to write an inequality for this set, letting x be any number in the group, we would say:

This same set could be described in another type of notation called interval notation. In that notation the group of numbers would be written as:

Here is how to interpret this notation: • •

•

•

The span of numbers included in the group is often imagined as being on a number line, usually the x-axis. The ‘(5’ on the left means the set of numbers starts at the real number which is immediately to the right of 5 on the number line. It means you should imagine a number the tinniest bit greater than 5, and that is where the group of numbers begins. The parenthesis to the left of 5 is called a round bracket or an exclusive bracket. That is, 5 is excluded from the group, but the numbers directly to the right of 5 are included. Simply put, numbers greater than 5 are included. The group of numbers continues to include values greater than 5 all the way to a value which is infinitely greater than 5. That is, the set of numbers goes all the way to positive infinity. That is what the positive infinity symbol on the right means. Infinity symbols are always accompanied by round brackets.

Now consider the group of numbers that are equal to 5 or greater than 5. That group would be described by this inequality:

In interval notation this set of numbers would look like this:

This interval notation would be interpreted just like the interval above, except:

•

The ‘[5’ on the left means the set of numbers starts on the number line with 5. The square bracket to the left of 5 is called an inclusive bracket. That is, 5 is included within the group. Simply put, the number 5 and all numbers greater than 5 are included.

Now, what about numbers greater than 5 but less than 7? Expressed as an inequality this group would look like this:

This same group of numbers expressed with interval notation would look like this:

Again the round, exclusive brackets on the left and right mean ‘up to but not including’. And here is an inequality showing a group of numbers equal to or greater than 5 and less than 7:

Here is this group of numbers expressed with interval notation:

Notice that there is a square, or inclusive, bracket on the left of this interval notation next to the 5. This means that this group of numbers starts at 5 and continues for values greater than 5. The round bracket on the right next to the 7 is, again, an exclusive bracket. This means that the numbers in this group have values up to but not including the 7. Well, by now, hopefully interval notation is clear to you. Let us go through one last simple example. Consider the group of numbers equal to or greater than 5 and less than or equal to 7. An inequality for this set would look like this;

Since both the 5 and the 7 are included in the group we will need inclusive, or square, brackets at each end of the interval notation. That notation looks like this:

Well, let us get just a bit more complicated. Using interval notation we will show the set of number that includes all real numbers except 5. First, stated as inequalities this group looks like this:

The statement using the inequalities above joined by the word or means that x is a number in the set we just described, and that you will find that number somewhere less than 5 or somewhere greater than 5 on the number line. In interval notation a logically equivalent statement does not use the word or, but rather a symbol for what is called the union of two groups of numbers. The symbol for union coincidentally looks like a U, the first letter of union. However, it is really not a letter of the alphabet. Here is what the union symbol looks like:

So, the group of numbers that includes all values less than 5 and all values greater than 5, but does not include 5 itself, expressed as interval notation looks like this:

Let us consider one last set of numbers. We will consider a group of numbers containing all numbers less than or equal to 5 and also those numbers that are greater than 7 but less than or equal to 12. Using inequalities this group of numbers could be notated like this:

And using interval notation as described throughout this material this group would look like this:

We would interpret this interval notation as representing the total group of numbers as the union of two other groups. The first would start at negative infinity and proceed toward the right down the number line up to and including 5. The second would start just to the right of 7, but not including 7, and continue to the right down the number line up to and including 12. The total set of numbers would be all those in the first group along with all of those in the second, and this would be the same total group of numbers which we considered in the above inequality where we first introduced this last example.

So, we see that interval notation is useful for stating the members of groups of numbers. It is often used to state the set of numbers which make up the domain and range of a function.

What is a perfect square? A perfect square is a quadratic expression that factors into two identical binomials. For example, if you multiply these binomials: (x + 2)(x + 2) You get: x2 + 4x + 4 Therefore, the quadratic expression x2 + 4x + 4 is a perfect square since it factors into two identical binomials which are (x + 2) and (x + 2). Notice that (x + 2)(x + 2) can be written (x + 2)2. So: x2 + 4x + 4 = (x + 2)2 Here are some more examples of perfect squares: x2 + 10x + 25 = (x + 5)(x + 5) = (x + 5)2 x2 - 6x + 9 = (x - 3)(x - 3) = (x - 3)2

Where does the quadratic formula come from? First of all, let us be sure we understand what is meant by the term quadratic formula. To do that, we must understand what is meant by the terms quadratic expression and quadratic equation. A quadratic expression is a polynomial of this form:

A quadratic equation is an equation of the form:

Examples of quadratic equations would be: 0 = 6x2 + 2x + 9 0 = 3.2x2 + 8.2x - 1.6 When equations of this form are solved for x, you get you get the quadratic formula, which is:

That is, you can use this calculation to find the values of x that make the quadratic equation equal to zero. These specific values of x are called the roots of the quadratic equation. They are also called the zeros of the equation. Let us figure out why x is equal to the above value. First, start with the quadratic formula:

[1] Subtract c from both sides to get:

[2] Divide both sides by a:

[3]

This next step may look a bit mysterious at first. It will allow us to turn the left side of the above equation into a perfect square. This method of turning one side of an equation into a perfect square is often called 'completing the square'.

Add

to both sides:

[4] Factor the left side of this equation by asking this question: What two numbers

when added are

That number is

and when multiplied are

?

.

Therefore, we can see the left side of [4] as a perfect square that factors this way:

[5] The left side can now be written this way:

[6] Let us do a bit of algebra on the right side of [6]:

[7] Now, we get a common denominator on the right side:

[8] Adding on the right side:

[9] Taking the square root of each side:

[10]

Subtract

from each side:

[11] Do some algebra on the right side:

[12]

Since the terms on the right side now have a common denominator, we will combine those terms by adding them:

[13] And we are all done. [13] is the solution to the quadratic equation.

Infinity and Its Symbol In mathematics an unimaginably large value is said to be at infinity or is said to be infinity. For the moment, let us consider such a value to be positive. That is, think of an x number line with the origin, or zero, on it. In your imagination have it extend from zero to the right forever. One could say that infinity is at the end of that line. Except that line has no end. It goes on without end. So, how far away to the right of zero is positive infinity? Well, if you started at the origin and spent your whole life running toward infinity at a million miles per hour, and somehow miraculously lived for a million lifetimes, at the end of all of that you would be no closer to infinity than when you started. That puts infinity very far away indeed. The symbol for this value looks like this:

This symbol is often read ‘positive infinity’. Just like for normal numbers, the positive sign on infinity can be implied or ‘invisible’. Therefore, this symbol has the same meaning:

What about the symbol for negative infinity, that value which is forever distant to the left of the origin? That symbol looks like this:

One could think of the positive infinity symbol as a variable with a value that is larger than any other positive number or that is less, in the case of negative infinity, than any negative number.

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