Name: ________________________ Class: ___________________ Date: __________
PreCalc 12 Chapter 2 Review Package
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. Use graphing technology to solve: x  1  x   10
Give the solution to the nearest tenth.
A. x 
Ö 7.5
B. x 
Ö 7.1
C. x 
Ö 13.9
D. x 
Ö 13.5
2. The graph of which function below has a hole?
A.
B.
x2
x2  2
x2  9
y
x3
y
C.
y
x2
x4
D.
y
x2  3
x2  2
3. State the domain of this rational function.
y
x 2  2x  8
x4
A. domain: y  6
B. domain: x  4
C. domain: x  4
D. domain: x  ò
1
ID: A
Name: ________________________
ID: A
4. The graph of which function below has a vertical asymptote at x  2 ?
A.
B.
x2
x4
x2
y
x2
y
x2  4
x2
x2
D. y  2
x 4
C.
y
5. The graph of which function below has a horizontal asymptote?
A.
y
x2  4
x7
B.
y
x 2  2x  8
x7
x2  5
x2  3
x2
D. y 
x4
C.
y
6. For the graph of this rational function, state the domain and write the equations of any asymptotes and the
coordinates of any hole.
y
2x  2
x  4x  3
2
A. domain: x  0 and x  1 ;
vertical asymptote: x  0 ;
hole: (1,1);
horizontal asymptote: y  3
B. domain: x  3 and x  1 ;
vertical asymptote: x  3 ;
hole: (1,1);
horizontal asymptote: y  0
C. domain: x  3 ;
vertical asymptote: x  3 ;
horizontal asymptote: y  1
D. domain: x  3 and x  1 ;
vertical asymptotes: x  3 , x  1 ;
horizontal asymptote: y  0
7. What is the equation of the vertical asymptote of the graph of this function?
x4
y 2
x  8x  16
A. x  4
B. x   4
C. x  0
D. The graph has no vertical asymptote.
2
Name: ________________________
ID: A
8. The graph of which function below has a horizontal asymptote?
A.
B.
x2  5
x7
x2
y
x4
y
x 2  9x  20
x7
x5
D. y  2
x 4
C.
y
9. For the graph of this rational function, state the domain and write the equations of any asymptotes.
y
3x 2  4
x 2  36
A. domain: x  3 ;
vertical asymptote: x  3 ;
horizontal asymptote: y  3
B. domain: x  3 ;
vertical asymptote: x  3 ;
horizontal asymptote: y  6
C. domain: x  6 ;
vertical asymptotes: x  6 ;
horizontal asymptote: y  6
D. domain: x  6 ;
vertical asymptotes: x  6 ;
horizontal asymptote: y  3
10. What is the solution of this radical equation, to the nearest tenth if necessary?
x8
5 x9
x9
A. x  8
B. x 
Ö 9.7 or x 
Ö 13.7
C. x  9
D. x 
Ö 13.7 or x 
Ö 9.7
11. State the domain of this function.
x2
y 2
x  6x  8
A. x  4 , x  2
B. x  4 , x  2
C. x  0
D. x  ò
3
Name: ________________________
ID: A
12. For the graph of this rational function, identify the equations of any asymptotes and the coordinates of any hole.
x4
y 2
x  8x  15
A. The graph has a vertical asymptote at x  5 , a hole at (3,7), and a horizontal asymptote at
y  0.
B. The graph has vertical asymptotes at x  5 and x  3 , and a horizontal asymptote at
y  0.
C. The graph has a vertical asymptote at x  0 , a hole at (3,7), and a horizontal asymptote at
y  5.
D. The graph has a vertical asymptote at x  0 , and a horizontal asymptote at y  5 .
13. Which function below is represented by this graph?
A.
y
3x 2  6x  4
x2
B.
y
3x  9
x2
3x 2  3x  18
x2  x  6
3x 2  9x  6
D. y 
x2
C.
y
14. For the graph of this rational function, identify the equation of any asymptote.
6x  8
y 2
x 4
A.
B.
C.
D.
The graph has an oblique asymptote at y  6x  8 .
The graph has a vertical asymptote at x  2 .
The graph has a horizontal asymptote at y  0 .
The graph has no vertical or horizontal asymptotes.
4
Name: ________________________
ID: A
15. For the graph of this rational function, identify the equations of any asymptotes and the coordinates of any hole.
x2
y 2
x  7x  10
A. The graph has a vertical asymptote at x  0 , and a horizontal asymptote at y  5 .
1
B. The graph has a vertical asymptote at x  5 , a hole at (2,  ), and a horizontal asymptote
3
at y  0 .
1
C. The graph has a vertical asymptote at x  5 , a hole at (2,  ), and a horizontal asymptote
3
at y  5 .
D. The graph has vertical asymptotes at x  5 and x  2 , and a horizontal asymptote at
y  0.
16. Which graph represents the function y 
x3  x
?
x2  1
A.
C.
B.
D.
5
Name: ________________________
ID: A
Short Answer
17. For the graph of y  f(x) shown below, sketch the graph of y 
f(x) .
18. For the graph of y  f(x) shown below, sketch the graph of y 
f(x) .
19. Use technology to graph this function, then complete the table below.
x 2  7x  12
y
x4
Vertical Asymptote(s)
Horizontal Asymptote
6
Non-permissible values of x
Name: ________________________
ID: A
20. Use technology to graph this function, then complete the table below.
x4
y 2
x  2x  8
Vertical Asymptote(s)
Horizontal Asymptote
Non-permissible values of x
21. Does the graph of this function have any holes or vertical asymptotes?
Write the coordinates of any holes and the equations of any vertical asymptotes.
5x 2  40x  60
y 2
x  2x  24
22. Does the graph of this function have an oblique asymptote?
If it does, write its equation.
x2
y
x3
Problem
23. For the graph of y  f(x) shown below, sketch the graph of y  f(x) .
State the domain and range of each function. If the domains and ranges are different, explain why.
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Name: ________________________
24. Given the graph of y  f(x) , sketch the graph of y 
What are the coordinates of the invariant points?
ID: A
f(x) .
3
25. Determine whether the graph of this function has an oblique asymptote. If it does, write its equation. Show your
work.
x2  x  6
y
x3
26. Two cars travel to Edmonton at the same time. Car A travels 240 km in the same time it takes Car B to travel 180
km. Car A travels an average of 25 km/h faster than Car B. A rational equation that relates the average speeds of
240
180

the two cars is:
, where v kilometres per hour is the average speed of Car B. Determine the average
v  25
v
speed of each car, to the nearest kilometre per hour.
27. Create an equation for a rational function whose graph is the line y  x with a single hole. Explain your thinking.
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