1. Lesson 2.7 Graphs of Rational Functions
Essential Question: How do you sketch the graph of the rational function 𝑓 𝑥 = 𝑁 𝑥 𝐷 𝑥 ?

2. Before we start…
Find the asymptotes and holes of the rational function.
𝑓 𝑥 = 3 𝑏 2 −20𝑏−7 𝑏 2 −14𝑏+49

3. What is a rational function?
A function that is a ratio of two polynomials where the denominator is not 0
𝑓 𝑥 = 𝑁 𝑥 𝐷 𝑥
𝑓 𝑥 = 𝑥 2 −6𝑥+5 𝑥 2 +10𝑥+21

4. How do you sketch the graph of the rational function 𝑓 𝑥 = 𝑁 𝑥 𝐷 𝑥 ?
To sketch the graph:
Simplify f, if possible.
Find and plot the y-intercept.
Find the x-intercepts by finding zeros of the numerator.
Find vertical asymptotes and holes (the zeros of the denominator).
Find and sketch any other asymptotes.
Plot at least one point between and one point beyond each x-intercepts.
Use smooth curves to complete the graph.

5. Library of Parent Functions: Rational Function
Graph of 𝑓 𝑥 = 1 𝑥
Domain: −∞, 0 ∪(0,∞)
Range: −∞, 0 ∪(0,∞)
No intercepts
Decreasing on −∞, 0
and (0,∞)
Odd function
Origin symmetry
Vertical asymptote: y-axis
Horizontal asymptote: x-axis

6. Sketch the graph of the function and describe how the graph is related to the graph of
𝑓 𝑥 = 1 𝑥 .
𝑔 𝑥 = 1 𝑥−3

7. Sketch the graph of the function and describe how the graph is related to the graph of
𝑓 𝑥 = 1 𝑥 .
ℎ 𝑥 = 1 𝑥+1 −2

8. Sketch the graph of 𝑓 𝑥 = 1 𝑥+3 and state its domain.

9. Sketch the graph of 𝐶 𝑥 = 2𝑥−1 𝑥 and state its domain.

10. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥 2 −𝑥−2 .

11. Sketch the graph of 𝑓 𝑥 = 𝑥 2 −4 𝑥 2 +4𝑥+4 .

12. Sketch the graph of 𝑓 𝑥 = 𝑥 2 −9 𝑥 2 −2𝑥−3 .

13. Slant Asymptotes
Consider a rational function whose denominator is of degree 1. If the degree of the numerator is exactly one more than the degree of the denominator, then the graph of the function has a slant (or oblique) asymptote.

14. How do you find a slant asymptote?
Use long division to divide the denominator into the numerator.

15. Sketch the graph of 𝑓 𝑥 = 𝑥 2 −𝑥 𝑥+1 .

16. Sketch the graph of 𝑓 𝑥 = 𝑥 2 −𝑥−2 𝑥−1 .

17. Sketch the graph of 𝑓 𝑥 = 3 𝑥 2 +1 𝑥 .

18. A rectangular page is designed to contain 48 square inches of print
A rectangular page is designed to contain 48 square inches of print. The margins on each side of the page are inches wide. The margins at the top and bottom are each 1 inch deep. What should the dimensions of the page be so that the minimum amount of paper is used?

19. How do you sketch the graph of the rational function 𝑓 𝑥 = 𝑁 𝑥 𝐷 𝑥 ?

20. Ticket Out the Door
Sketch the graph of the rational function 𝑓 𝑥 = 𝑥+3 𝑥 2 −9