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.