4.3.4.4.CPMath.Rational.Functions.pdf

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College Prep Math ‐ Section 4.3 & 4.4 ‐ Rational Functions

Starter Questions‐ Which fractions are proper and which are improper? a. b. c.

State the degree of each polynomial. d. 3 e. 1 f. 7

The root word of the word rational is ___________. A ratio is a fraction; it’s a division problem. In these sections we

will learn about rational functions and how to graph them. To get the graphs correct we will need to remember basic

graphing skills, how to find the domain of a function, factoring, simplifying, and various other algebra and graphing

skills. In other words, there is a lot to remember and keep track of. The problems are not hard, but we will need to

do them in a very systematic way and think about each piece of information we get and what it means for the graph.

There is a lot of logic that can be learned and applied when dealing with rational functions.

A Rational Function is of the form: ,

where and are __________ and is not the _____ polynomial.

The domain of a rational function is the set of all real numbers except those that make equal to ________.

Why can’t be equal to zero?

Let’s look at some very simple and familiar rational functions.

Example 1‐ Graph and then identify the asymptotes, domains, ranges, and the intercepts.

Two of the most useful vocabulary words that can be very helpful when dealing with rational functions are:

Proper‐ The degree of the numerator is lower than the degree of the denominator.

Improper‐ The degree of the numerator is _________ to or _________ than the degree of the denominator

Example 2‐ Determine whether the rational function is Proper or Improper.

a.

b.

c.

When graphing rational functions we use proper and improper to help us find the horizontal and oblique

asymptotes. (Oblique means ____________.)

1. If the rational function is proper, then the horizontal asymptote is . 2. There are two possible outcomes if the rational function is improper:

“Good Type” of Improper‐ The degree of the top is the same as the degree of the bottom.

The horizontal asymptote is .

“Bad Type” of Improper ‐ The degree of the top is higher than the degree of the bottom. If this is the case, use long division, then is the

asymptote. (This usually produces an ___________ asymptote.)

Does the graph for example 1 fit the descriptions given for proper and improper rational functions? 1

2

What would happen to the graphs if

the functions changed to:

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Example 3‐ Find the horizontal or oblique asymptote for each of the following.

a.

b.

c.

After finding the horizontal and oblique asymptotes, we need to find the vertical asymptotes and any “holes” in the

graph. We usually find vertical asymptotes and holes in the graph by first finding the zeros of the denominator. We

do this by factoring the numerator and denominator. A number that makes the denominator equal to zero is not in

the domain. (That’s why we call it a zero of the denominator.) If the factor that produced the zero cancels, there is a

hole in the graph at that value. If the zero doesn’t cancel, there is a vertical asymptote at that value.

Example 4‐ Find the vertical asymptotes and/or holes for each of the following and state the domain.

a.

b.

c.

Finding the asymptotes and holes is a major part of understanding rational functions and their graphs. However, we

can’t forget all the other important things we learned about graphs like: intercepts, end behavior, multiplicity, etc.

Example 5‐ Find the ‐intercepts (i.e. zeros) of each rational function.

a.

b.

c.

*Only the _____________ can produce ‐intercepts.

Sometimes you’ll be given an easy problem where all you have to do is identify features from a graph of a rational

function.

Example 6‐ Use the graph to find:

a. The domain and range.

b. The intercepts.

c. Horizontal asymptotes, if any.

d. Vertical asymptotes, if any.

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Now that we can find and identify important features on the graph of a rational function, it’s time to put all of this

together so we can produce the graph of a rational function.

How

to

Graph

a

Rational

Function

1. Locate the horizontal or slanted asymptotes by determining if the function is proper or improper.

If Proper, then

If Improper: Good Type, then

Bad Type‐(use long division), then

2. Factor the denominator to find the domain, then factor the numerator and cancel any common factors.

3. Locate: vertical asymptotes ‐ (zeros in the denominator that don’t cancel)

holes in the graph‐ (zeros in the denominator that cancel)

4. Find the and ‐intercepts. (Remember, only the numerator can make a fraction equal zero.)

5. Draw the asymptotes, holes, and intercepts on the graph. Then use an , ‐table to determine where the

function is above (pos.) or below (neg.) the axis and to determine the graphs behavior near asymptotes.

6. Put all the information together to draw a sketch of the graph. (Think about what each piece of information tells

you. What is an ‐intercept? What is an asymptote? What does multiplicity mean about the ‐intercept? Etc.)

7. Label important features. (If the steps above are done correctly, this is done automatically.)

Most people find it helpful to plot the each piece of information as they find it. However, you can do it any way you

like as long as you show all your work, label all the parts, and get it right.

Example 7‐ Graph the following rational function. Show your work and label all the important parts.

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Example 8‐

Thought Questions‐ Can a graph touch a vertical asymptote? Why or why not? Can a graph touch a horizontal asymptote? Why or why not?

Example 9‐ Graph the rational function

.

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.

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Thought Question‐ Why can we “ignore” the remainder on an improper rational function when finding the oblique

asymptote?

.