Unit 3, Lesson 3
Graphs of Rational Functions (Part 2)
Algebra 2
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Let be the function that gives the average cost per book , in dollars, when using an online store to print copies of a self-published paperback book. Here is a graph of
Here are four graphs of rational functions.
AGraph of a rational function f(x) with a dashed horizontal asymptote through (0 comma 4), on xy-plane. Each axis from -10 to 8, by 2’s. Vertical asymptote of the function at x = 0.
BGraph of a rational function f(x) with a dashed horizontal asymptote through (0 comma -4), on xy-plane. Each axis from -10 to 8, by 2’s. Vertical asymptote of the function at x = 0.
CGraph of a rational function f(x) with a dashed horizontal asymptote through (0 comma 1), on xy-plane. Each axis from -10 to 8, by 2’s. Vertical asymptote of the function at x = 0.
DGraph of a rational function f(x) with a dashed horizontal asymptote through (0 comma -1), on xy-plane. Each axis from -10 to 8, by 2’s. Vertical asymptote of the function at x = 0.
Consider the rational function . Written this way, we can tell that the graph of the function has a vertical asymptote at by reading the denominator and identifying the value that would cause division by 0. But what can we tell about the value of for values of far away from the vertical asymptote?
One way we can think about these values is to rewrite the expression for by breaking up the fraction:
Written this way, it’s easier to see that as gets larger and larger in either the positive or negative direction, the 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. Here is a graph of showing values from -40 to 40.
A dashed line at is included to show how the function approaches this value as inputs are farther and farther from . This is an example of a feature of rational functions: a horizontal asymptote.
The line is a horizontal asymptote for a function if the value of the function gets closer and closer to as the magnitude of increases.
More generally, if a rational function can be rewritten as , where is a constant and and are polynomial expressions in which gets closer and closer to 0 as gets larger and larger in both the positive and negative directions, then will get closer and closer to .
The line is a horizontal asymptote of a function if the outputs of the function get closer and closer to 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 -axis.