Rational Functions

When you are done with this section, you will be able to do the following

Find the domain of a rational function.
Find the vertical and horizontal asymptotes of the graph of a rational function.
Sketch the graph of a rational function.
Use a rational function model to solve an application problem.

We are now moving on to a topic that is related to polynomials in that a rational function is a function that has a numerator which is a polynomial and a denominator which is also a polynomial.

Right now the best place to find information about rational functions is your text book (pages 354-354 in College Algebra 3rd edition by Stewart, Redlin, and Watson).

Read carefully:

x-intercepts are found in the same way we found x-intercepts before (set the function equal to 0 and solve). All x-intercepts are of the form (x, 0)

y-intercepts are found by plugging in 0 for x and solving. All y-intercepts are of the form (0, y).

We have found domains of rational functions previously. We just need to eliminate where the denominator is equal to 0. Domains in WeBWorK must be written in interval notation.

Vertical asymptotes are lines (x = a) that the function cannot cross. As the function approaches the vertical asymptote its y values go either toward infinity or negative infinity.

A horizontal asymptote is a line (y = b) that the function can cross. It is a value that the function approaches as the x values get very large (approach infinity) or as the x values get very small (approach negative infinity).

You will basically need to be able to find each of these for any rational function.

Additional Rational Functions Resources On-line Resources:

Please note that you may read about holes, oblique asymptotes, and singularities in rational functions. For this particular course, we will not discuss those topics. You may find that you run into them in a later math course.

Rational Functions Homework:

The Rational Functions Homework contains 5 problems.
Things to remember:

Many of the word problems ask you to find "future value" for a given model. What you are looking for in the models is the Horizontal Asymptote. If a rational function has an horizontal asymptote, it approaches a limiting value. This is what the function approaches as the input values approaches positive infinity or negative infinity. (increases or decreases without bound).

Make sure you give your answers in correct units. i.e. some of them the output might be stated to be in millions, so if you get 6 as your answer as an output from the model, you need to enter in 6,000,000 into WebWork as the solution.

Domains must be written as intervals. [ and ] indicate that the endpoint is included. ( and ) indicate that the endpoint is not included. Click here for a review of interval notation.

Intercepts are points and must be written as ordered pairs (x-intercepts look like (x, 0) and y-intercepts look like (0, y)).

All asymptotes are lines and must be written as equations. Vertical lines must be written as x = a. Horizontal lines must be written as y = b.

Homework examples:

Example 1: similar to problem 1

Example 2: similar to problem 2

Example 3: similar to problem 3

Keep in mind that it is recommended that you also complete all the problems in the set called "also recommended" with the same number as the Rational Functions Homework. There are many different ways to ask the same questions. This will allow you to see additional problems that are related to this topic.