1. Rational Functions and Asymptotes
Section 2.6
Rational Functions and Asymptotes

2. Objective By following instructions students will be able to:
Find domains of rational functions.
Find horizontal and vertical asymptotes of graphs of rational functions.
Use rational functions to model and solve real-life problems.

3. Rational Functions Vertical asymptote x=0
Vertical asymptotes: “certain” asymptote
D(x)=0
(solve for the variable)
Horizontal asymptote
y=0

4. Rational Functions
Horizontal asymptotes “uncertain” asymptote (determined by degree) n=degree of N(x) d=degree of D(x)
Less Than
Equal to
Greater Than
n=d +1
no horizontal asymptote

5. Example 1:
Find the horizontal asymptote of each function. Use a graphing calculator to check your work. a) b) c)

6. Example 2:
Find the domain of and discuss the behavior of the Function near any excluded values.

7. Example 3:
Graph . Identify the domain and the range. Discuss the behavior of the function near any excluded values.

8. Example 4: Find the domain Vertical asymptotes Horizontal asymptotes
Zeros, if any.

9. U-TRY #1
(a) Find the domain (b) Identify any horizontal and vertical asymptotes (c) verify your answers by creating a table. 1) 2)

10. Example 5:
A utility company burns coal to generate electricity. The cost of removing a certain percent of the pollutants from the smokestack emissions is typically not a linear function. That is, if it costs C dollars to remove 25% of the pollutants, it would cost more than 2C dollars to remove 50% of the pollutants. As the percent of removed pollutants approaches 100%, the cost tends to become prohibitive. Suppose that the cost C (in dollars) of removing p% of the smokestack pollutant is Sketch the graph of this function. Suppose you are a member of a state legislature that is considering a law that would require utility companies to remove 90% of the pollutants from their smokestack emissions. If the current law requires 85% of the removal, how much additional cost would there be to the utility company because of the new law?

11. Example 6:
A business has a cost function of , where C is measured in dollars and x is the number of units produced. The average cost per unit is Find the average cost per unit when x=1000, 5000, 10,000, and 100,000. What is the horizontal asymptote for this function, and what does it represent?

12. Revisit Objective Did we… Find domains of rational functions?
Find horizontal and vertical asymptotes of graphs of rational functions?
Use rational functions to model and solve real-life problems?

13. Homework
Pg 195 #s 1-31 ODD