Algebra 2
Rational Functions Test Review
Name:
Pd:
Work out answers on another sheet of paper and attach to this sheet. Box or circle all answers. This review is
due on the day of the test and will be a quiz grade. No work shown = No credit.
1. t varies directly with the square of x, and inversely with b and s. Find the expression relating the
variables if when x = 15, b = 6, and s = 5, then t =15. Then find x when t = 24, b = 3, and s = 4.
2. m varies jointly with p and s, and inversely with the square of c. Find the expression relating the
variables if when m = 1/3, p = 3, s = 2, and c = 3. Then find c when m = 3, p = 15, and s = 10.
3. Determine if the following vary directly, inversely, or neither:
1
x
1
a. 3 x  y
b. xy  1
c.
d. 2 x  5  y e.
x

4 y 12
3y
4. Determine if the following vary directly, inversely, or neither:
a.
b.
c.
x y
x y
2
6
2
70
5
15
5
28
7
21
7
20
x
y
2
28
5
13.2
7
8
Graph the following transformations of the rational function’s parent function.
5. y =
1
x3
6. y =
1
3
x
7. y =
1
4
x2
8. y =
1
1
x5
9. y =
3
2
x 1
Find the important information (including all intercepts, asymptotes and crossings, holes, domain, and range) and
graph the following. (Review from last semester: find the inverse function of 10-12).
10. y =
x3
x2
13. y =
3x  3
x 1
16. y =
4x 2  5x  6
3x 2  4 x  4
11. y =
x4
2x 1
14. y =
x 8
x2  4
17. y =
2 x 2  3x  9
x 2  4x  5
12. y =
2x  8
x4
15. y =
2x 2  x 1
x2  4
18. y =
3x 2  x  4
3x 3  2 x 2  3x  2
Algebra 2
19. y =
15 x 2  2 x  8
4 x 3  5x 2  2 x  3
20. y =
4 x 3  3 x 2  16 x  12
3 x 2  10 x  8
21. y 
2 x 3  x 2  16 x  15
4 x 4  12 x 3  5 x 2  12 x  9
Solve the following Rational Equations.
22.
3
1 1
 
x  2 x 5x
23.
10
4
5
 
x  2x x x  2
24.
2
x
1
2

 2
x  2 x  4 x  6x  8
Simplify the following rational expressions.
25.
x 1 2x  5

x3 x2
27.
x
1
3

 2
x  2 2x  4 x  2x
26.
2
8
 2
x  25 x  10 x  25
2
1
x
28.
2
3 2
x
4
1
 x3
29. x  1
1
x 3
x4
y x

x y
30.
x y
xy
Solve the rational inequality, graph the solution on a number line, and write your solution in interval notation:
31.
2x  7
3
x5
32.
 3x
1
x2  4
Algebra 2
33. The parks and wildlife division of a county government introduces 80,000 fish into an artificial lake.
204  3t 
The population of the fish (in thousands) is represented by the equation P t  
, where t is the
1  0.05t
elapsed time in years ( t ≥ 0). (a) Graph the population function for the fish, and find (b) the population
after 10 years, (c) the year in which the population reaches 500,000 fish, and (d) the theoretical
maximum number of fish in the pond (explain why).
34. A bear shows up in the county park, attracted by the new fishing opportunities. The rangers decide to
remove the bear by sedating it with a tranquilizer dart. The concentration of sedative (in grams per liter)
3t 2  t
in the bear’s bloodstream t minutes after being darted is C t   3
.
t  64
a) Using a graphing calculator, determine the maximum concentration of sedative in the bear’s
bloodstream and when it will occur, correct to the nearest hundredth.
b) If the bear will be knocked out when the concentration of the sedative in its bloodstream rises above
0.2 grams per liter, and will wake up when it falls below that level again, how long does the ranger
team have to safely remove the unconscious bear from the property, rounded to the nearest tenth of a
minute?
35. A package in the shape of a rectangular prism with a square base needs to be wrapped. The volume of
the package is 2880 cu.in.
a) Write a function that gives the height h of the rectangular prism in terms of the side s of the base.
b) Write an equation for the surface area of the rectangular prism just in terms of s, and use your
calculator to find the value of s that minimizes the amount of wrapping paper required for the
package.
Algebra 2
Key:
2x 2
, x = 12
bs
ps
2. m  2 , c = 5
2c
3. a. directly
1. t 
b. inversely
c. inversely
d. neither
e. directly
4. a. directly
b. inversely
c. neither
10. VA x = -2, HA y = 1, x-int (3, 0), y-int (0, -3/2), D: (-∞, -2)U[-2, +∞), inverse: y =
 2x  3
x 1
11. VA x = -½, HA y = ½, x-int (-4, 0), y-int (0, 4), D: (-∞, -½)U[-½, +∞), inverse: y =
4 x
2x 1
12. VA x = -4, HA y = 2,x-int (4, 0), y-int (0, -2), D: (-∞, -4)U[-4, +∞), inverse: y =
 4x  8
x2
13. No VA, HA, or x-int; y-int (0, 3), hole at (1, 3), D: (-∞, 1)U[1, +∞)
14. No VA, HA y = 0, x-int (8, 0), y-int (0, -2), crossing (8, 0), D: all real #s
15. VA x = -2 & x = 2, HA y = 2, x-int (0, -½) & (0, 1), y-int (0, ¼), crossing (7, 2), D: (-∞, -2)U(-2, 2)U[2, +∞)
16. VA x = -2/3, HA y = 4/3, x-int (0, -3/4), y-int (0, 3/2), hole (2, 11/8), D: (-∞, -2/3)U(-2/3, 2)U[2, +∞)
17. VA x = -5 & x = 1, HA y = 2, x-int (-3/2, 0) & (3, 0), y-int (0, 9/5), crossing (1/11, 2), D: (-∞, -5)U(-5, 1)U[1, +∞)
18. VA x = -1 & x = 2/3, HA y = 0, x-int (-4/3, 0), y-int (0, -2), hole (1, 7/2), crossing (-4/3, 0), D: (-∞, -1)U(-1,
2/3)U[2/3, +∞)
19. VA x = -3/4 & x = 1, HA y = 0, x-int (-4/5, 0) & (2/3, 0), y-int (0, -8/3), crossing (-4/5, 0) & (2/3, 0), D: (-∞, -3/4)U(3/4, 1)U(1, +∞)
20. VA x = -4/3, no HA, OA y = 4/3 x – 31/9, x-int (-3/4, 0) & (2, 0), y-int (0, -3/2), hole (-2, -10), D: (-∞, -2)U(-2, 4/3)U(-4/3, +∞)
21. VA x = -3/2 & x = 1; HA y = 0, x-int (-5/2, 0) & (3, 0), y-int (0, 5/3), hole (-1, 6), crossings (-5/2, 0), D: (-∞, -3/2)U(3/2, -1)U(-1, 1)U(1, +∞)
Algebra 2
(10-21 graphs at end)
22. 4/3
23. no solution
24. -1
3x 2  10 x  13
25.
( x  3)( x  2)
26.
 6 x  45
( x  5) 2 ( x  5)
27.
2x  3
2x
28.
4x 2  x
, x ≠0
3x 2  2
29.
 1 3 5
x 3  6x 2  6x  8
, x ≠ 1, -4,
3
2
x  12 x  11
30. y  x , x ≠ 0, y ≠ 0, and x ≠ y
31. (-∞, 5)U[8, +∞)
32. [-4, -2)U[1, 2)
33. (a)
(b) about 453 fish
(c) 12 years
(d) 1,200,000 fish; the function has a horizontal asymptote there, so it is approaching that value.
34. (a) Maximum: 0.424 g/L; it happens at 4.884 minutes
(b) 13.0 minutes, so they’d better move fast!
35. (a) hs  
2880
s2
(b) SA  2 s 2 
11520
; min. surface area is 1213.543 cm2 and it occurs when s = 14.228 cm
s
Algebra 2
Graphs (this program doesn’t show arrows but you should):
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Algebra 2
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Algebra 2
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Algebra 2
19)
20)
21)
Algebra 2
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Algebra 2 Rational Functions Test Review Name: Pd: Work out