Secondary Math 3 H - Section 2.6 - Rational Functions
Starter Questions- Which fractions are proper and which are improper? a. b. c.
State the degree of each polynomial. d. e. f.
The root word of the word rational is _________. A ratio is a fraction; it’s a _____________ problem. To get the graphs of a rational function 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 the simplest rational function.
Example 1- Graph then identify the domain, range, and the intercepts.
Most rational functions have asymptotes. An asymptote is __________ or a curve that approximates the graph. Asymptotes help us get the ___________ of the graph right.
Vertical Asymptote (VA)- A vertical line that the graph ______________ either from the left or the right.
(A vertical asymptote, it forces the graph to either go up or down when it approaches the vertical asymptote from either side.)
Horizontal Asymptote (HA)- A horizontal line that the graph approaches on the right or left end of the graphs.
(A horizontal asymptote forces the graph to approach the horizontal line as approaches or , the left end or right end.)
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.)
What would happen to the graphs if the functions changed to:
Does the graph for example 1 fit the descriptions given for proper and improper rational functions?
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. A “hole” is exactly what is sounds like, it is a single missing point on the graph. We usually find vertical asymptotes and holes in the graph by first finding the zeros of the ___________. We do this by factoring the numerator and denominator. A number that makes the denominator equal to zero is not in the domain. If the factor that produced the zero cancels, there is a ________ 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 of each rational function. (*Be sure to use the simplified form to find -intercepts.)
a.
b.
*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 and match it with function that could produce that graph.
Example 6- Match the function to the graph. i.
ii.
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 ____________________ 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 ___________. 3. Locate: vertical asymptotes - (zeros in the denominator that DO NOT __________)
holesin the graph- (zeros in the denominator that do ____________)
4. Find the and -intercepts. (Remember to use the ________________ form and only the ______________ 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. Plot these “extra” points.
6. Put all the information together to draw a sketch of the graph. (Think about what each piece of information tells you about the graph. 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 step 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.
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 10- Graph the rational function
.
Example 11- Graph the rational function
.
Example 12- Graph the rational function
