Lesson 18

Graphs of Rational Functions (Part 2)

  • Let’s learn about horizontal asymptotes.

18.1: Rewritten Equations

Decide if each of these equations is true or false for \(x\) values that do not result in a denominator of 0. Be prepared to explain your reasoning.

  1. \(\displaystyle{\frac{x+7}{x}=1+\frac{7}{x}}\)
  2. \(\displaystyle{\frac{x}{x+7} =1+\frac{x}{7}}\)

18.2: Publishing a Paperback

Let \(c\) be the function that gives the average cost per book \(c(x)\), in dollars, when using an online store to print \(x\) copies of a self-published paperback book. Here is a graph of \(c(x)= \tfrac{120+4x}{x}.\)

Coordinate plane, total books printed, 0 to 500 by 50, cost per book, 0 to 20 by 1. Curve beings near 1 comma 20 and drops steeply toward 50 comma 6 point 2, then gently toward 500 comma 4 point 1.
  1. What is the approximate cost per book when 50 books are printed? 100 books?
  2. The author plans to charge \$8 per book. About how many should be printed to make a profit?
  3. What is the value of \(c(x)\) when \(x=\frac{1}{2}\)? How does this relate to the context?
  4. What does the end behavior of the function say about the context?

18.3: Horizontal Asymptotes

Here are four graphs of rational functions.

A

Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.

B

Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.

C

Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.

D

Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.

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  1. Match each function with its graphical representation.
    1. \(a(x)=\frac{4}{x}-1\)
    2. \(b(x)=\frac{1}{x}-4\)
    3. \(c(x)=\frac{1+4x}{x}\)
    4. \(d(x)=\frac{x+4}{x}\)
    5. \(e(x)=\frac{1-4x}{x}\)
    6. \(f(x)=\frac{4-x}{x}\)
    7. \(g(x)=1+\frac{4}{x}\)
    8. \(h(x)=\frac{1}{x}+4\)
  2. Where do you see the horizontal asymptote of the graph in the expressions for the functions?


Consider the function \(a(x) = \frac{\frac{1}{2}x+1}{x-1}\).

  1. Predict where you think the vertical and horizontal asymptotes of \(a(x)\) will be. Explain your reasoning.
  2. Use graphing technology to check your prediction.

Summary

Consider the rational function \(f(x) = \frac{3x+1}{x}\). Written this way, we can tell that the graph of the function has a vertical asymptote at \(x=0\) by reading the denominator and identifying the value that would cause division by zero. But what can we tell about the value of \(f(x)\) for values of \(x\) far away from the vertical asymptote?

One way we can think about these values is to rewrite the expression for \(f(x)\) by breaking up the fraction:

\(f(x) = \frac{3x}{x} + \frac{1}{x} \\ f(x)= 3 + \frac{1}{x}\)

Written this way, it’s easier to see that as \(x\) gets larger and larger in either the positive or negative direction, the \(\frac{1}{x}\) term will get closer and closer to 0. Because of this, we can say that the value of the function will get closer and closer to 3.

More generally, if a rational function \(g(x) = \frac{a(x)}{b(x)}\) can be rewritten as \(g(x) = c + \frac{r(x)}{b(x)}\), where \(c\) is a constant, and \(r(x)\) and \(b(x)\) are polynomial expressions where \(\frac{r(x)}{b(x)}\) gets closer and closer to zero as \(x\) gets larger and larger in both the positive and negative directions, then \(g(x)\) will get closer and closer to \(c\).

Rational functions of this type have a horizontal asymptote at the constant value. The line \(y=c\)  is a horizontal asymptote for \(f\) if \(f(x)\) gets closer and closer to \(c\) as the magnitude of \(x\) increases.

Glossary Entries

  • horizontal asymptote

    The line \(y =c\) is a horizontal asymptote of a function if the outputs of the function get closer and closer to \(c\) as the inputs get larger and larger in either the positive or negative direction. This means the graph gets closer and closer to the line as you move to the right or left along the \(x\)-axis.

  • rational function

    A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.

  • vertical asymptote

    The line \(x=a\) is a vertical asymptote for a function \(f\) if \(f\) is undefined at \(x=a\) and its outputs get larger and larger in the negative or positive direction when \(x\) gets closer and closer to \(a\) on each side of the line. This means the graph goes off in the vertical direction on either side of the line.