Name: _________________________________________________________ Unit 4 Review

IB SL YR 1 Pd. ______ Date: __________

Unit 4 : Rational Functions

4-1 Pre-requisite skills. Inverse functions Graphing x= and Y= lines. Rational functions Ratio of two polynomials.

4-2 X and y intercepts on a function. 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!)

4-3 Asymptotes 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 .

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.

Don’t forget the Rhyme! High up top nothing makes it stop High down low y equals zero.

4-4 Hole in the graph

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.

Procedure:1. Factorize top and bottom.2. Common factor equal to zero.3. Find other Vertical asymptotes.4. Check if the other vertical asymptote is also a hole.

DON’T FORGET! IF THERE IS AHOLE AT A PARTICULAR X VALUE, THERE CAN BE NO VERTICAL ASYMPTOTE OR X INTERCEPT THERE!

Station 1: Inverse and composition of functions that involve rational functions

1. 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.

2.

3. 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.

Station 2: Identifying Radical function graphs:

In questions 4 – 8, match each rational functions with its graph.

4. Graph:_____

5. Graph:_____

6. Graph:_____

7. Graph:_____

8. Graph:_____

Station 3: Identifying different parts of a rational function.:

For each function given, find the following (a) vertical asymptote(s) or hole(s), (b) horizontal asymptote, (c) y-intercept, (d) x-intercept(s), (e) Domain.

1.

1.

vertical asymptote(s) _________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

2.

vertical asymptote(s) _________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

Domain:__________________

x-intercept(s) _____________

Domain:_________________

3.

1.

vertical asymptote(s) _________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

4.

vertical asymptote(s) _________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

Domain:__________________ Domain:_________________

5.

1.

vertical asymptote(s) _________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

6.

vertical asymptote(s) _________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

Domain:__________________ Domain:_________________

7. 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?

8. 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 ?

9. The graph of the rational function f ( x )= axx−c is shown below. The graph has a vertical asymptote at

x=1 and a horizontal asymptote at y=3.

a. State the value of c.

b. Find the value of a.

10. The graph of the rational function f ( x )= qx−p is shown below. The graph has a vertical asymptote

at x=−2 and a horizontal asymptote at y=0. The poiny ( x , y )=(0 ,−12 )

a. State the value of p.

b. Find the value of q.