Name_________________________IB Math SL

Unit 5- Rational Functions Review

Answer the Looking Forward Goal Problem!

Analyzing the graph of a rational function

To find the vertical asymptotes of the graph of a function f , set the denominator of the function

equal to 0 and solve. For each value of x=c found, if f ( c )=non−zero ¿ ¿0 then the graph of f

has a vertical asymptote at x=c ,

If f ( c )=00 , then the graph of f has a hole at x=c . Any value of x that makes the denominator

equal zero should be excluded from the domain.

To find the horizontal asymptotes of the graph of a function f , use these guidelines.1. If the degree of the numerator is less than the degree of the denominator, then the graph of f has a horizontal asymptote at y=0.

2. If the degree of the numerator is equal to the degree of the denominator, then the graph of f has a horizontal asymptote given by the ratio of the leading coefficients.

3. If the degree of the numerator is greater than the degree of the denominator, then the graph of f does not have a horizontal asymptote.

Remember the Rhyme! High up top nothing makes it stop High down low y equals zero.

To find the y-intercept of the graph of a function f , find f (0).

To find the x-intercept(s) of the graph of a function f , set f ( x )=0. (Factor first!)

For each function given, find the following (a) domain (b) vertical asymptote(s) or hole(s), (c) horizontal asymptote, (c) y-intercept, (d) x-intercept(s). Then use your calculator to sketch the graph of the function.

1.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

2.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

3.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

4.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

5.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

6.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

7.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

8.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

9.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

10.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

11.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

12.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

13.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

14.

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

15. Consider the function: f ( x )= −4x−4

+6

a. Identify where the graph of the function intersects the axes.

b. What are the equations of any vertical asymptotes?

c. Are there any holes in the graph of f ?

d. What are the equations of any horizontal asymptotes?

e. What is the domain of f ?

In questions 16 – 23, match each rational functions with its graph.

16.

17.

18.

19.

20.

21.

22.

23.

24. Consider the function: f ( x )= 2x−2

+3

a. Identify where the graph of the function intersects the axes.

b. What are the equations of any vertical asymptotes?

c. Are there any holes in the graph of f ?

d. What are the equations of any horizontal asymptotes?

e. What is the domain of f ?

25. Let g ( x )=2 x−1 ,h ( x )= 3xx−2

, x≠ 2.

a. Find an expression for (h∘g )(x). Simplify your answer.

b. Solve the equation (h∘g ) ( x )=0.

26.

27. Consider the functions f : x 4(x – 1) and g : x 2–6 x

.

(a) Find g–1.

(b) Solve the equation ( f ° g–1) (x) = 4.