Precalculus
Review 1
Graph the function.
1
1) f(x) = |x|
x
Find the vertical asymptotes of the rational function.
x + 6
8) R(x) = x2 - 9
if 0 ≤ x < 3
if 3 ≤ x < 9
if 9 ≤ x ≤ 12
10
y
Give the equation of the oblique asymptote, if any, of the
function.
x2 + 9x - 5
9) F(x) = x - 5
5
-10
-5
5
10
15
x
-5
Give the equation of the horizontal asymptote, if any, of
the function.
5x2 + 6
10) F(x) = 5x2 - 6
-10
List the potential rational zeros of the polynomial
function. Do not find the zeros.
2) f(x) = 11x4 - x2 + 7
Graph the function.
x
11) f(x) = 2
x - 16
y
10
Use the Rational Zeros Theorem to find all the real zeros
of the polynomial function. Use the zeros to factor f over
the real numbers.
3) f(x) = x4 + 12x2 - 64
Information is given about a polynomial f(x) whose
coefficients are real numbers. Find the remaining zeros of
f.
4) Degree 4; zeros: 2 - 5i, 4i
Form a polynomial f(x) with real coefficients having the
given degree and zeros.
5) Degree 3: zeros: 1 + i and -8
Use the given zero to find the remaining zeros of the
function.
6) f(x) = x4 - 45x2 - 196; zero: -2i
Find the domain of the rational function.
x + 4
7) R(x) = x4 - 64
5
-10
-5
5
10
x
-5
-10
Find the indicated quantity.
12) If w = 6i + 3j, find 2w.
Find the unit vector having the same direction as v.
13) v = -4i + 3j
Solve the problem.
14) Find a vector v whose magnitude is 29 and
whose component in the i direction is
three times the component in the j direction.
Find the dot product v · w.
15) v = i + 4j,
w = 8i - j
Find the angle between v and w. Round your answer to
one decimal place, if necessary.
16) v = 2i - 3j,
w = 9i - 3j
Solve the problem.
17) Which of the following vectors is parallel to v
= i - j?
State whether the vectors are parallel, orthogonal, or
neither.
18) v = 4i + j,
w = i - 4j
Decompose v into two vectors v1 and v2 , where v1 is
parallel to w and v2 is orthogonal to w.
19) v = i + 7j,
w = i + j
Solve the problem. Round your answer to the nearest
tenth.
20) Find the work done by a force of 2 pounds
acting in the direction of 36° to the horizontal
in moving an object 7 feet from (0, 0) to (7, 0).
Solve the system of equations. If the system has no
solution, say that it is inconsistent.
26)
x - y + 2z = -1
3x + z = 0
-x + y - 2z = 2
Solve the system of equations.
27)
x + 4y - z = 3
x + 5y - 2z = 5
3x + 12y - 3z = 9
Solve the system of equations using substitution.
28)
x2 - y 2 = 39
x - y = 3
Solve using elimination.
29)
x2 + y 2 = 4
x2 - y 2 = 4
Solve the system of equations by substitution.
21)
x + 7y = -2
3x + y = 34
Solve the system of equations by elimination.
22)
2x + 5y = -3
2x + 2y = 12
Solve the system of equations. If the system has no
solution, say that it is inconsistent.
23)
8x - 9y = -2
8x - 9y = -5
Write the partial fraction decomposition of the rational
expression.
x
30)
(x - 6)(x - 7)
31)
45 - 9x
3
x - 6x2 + 9x
32)
14x + 1
(x - 1)(x2 + x + 1)
33)
x2 + 5x - 3
(x2 + 3)2
34)
3x3 + 9x - 4
(x2 + 2)2
35)
13x + 2
x3 - 1
24)
9x + y = 3
-45x - 5y = -15
Solve the system of equations.
25)
x - y + 4z = -21
4x + z = -4
x + 3y + z = 11
36)
6x2 - 14x + 22
(x - 2)(x2 + 5)
37)
7x3 - 2
x2 (x + 1)3
38)
39)
47)
x + y + z = 2
x - y + 5z = 18
4x + y + z = 5
48)
x + y + z = 7
x - y + 2z = 7
5x + y + z = 11
x
x2 + 5x + 6
7x2 - x - 20
x(x + 1)(x - 1)
Solve the problem.
49) Find real numbers a, b, and c such that the
graph of the function y = ax2 + bx + c contains
the points (1, 1), (2, 4), and (-3, 29).
2x - 5
40)
x2 - 5x - 6
41)
50) Find real numbers a, b, and c such that the
graph of the function y = ax2 + bx + c contains
the points (1, 2), (2, 11), and (-3, -14).
9x2 + 26x + 24
x(x + 6)(x - 2)
Solve using elimination.
42)
2x 2 + y 2 = 17
3x2 - 2y2 = -6
Solve the system of equations using substitution.
43)
ln x = 3ln y
3 x = 27y
Solve the system of equations by elimination.
51)
3x + 6y = 24
3x + 2y = 32
52)
3x - 5y = -12
6x + 8y = -24
53)
7
x - y = 10
12
5
x + 2y = 11
9
44)
y = x + 2
y2 = 8x
45)
5x2 + 9y2 = 81
y = x + 3
Solve the system of equations by substitution.
54)
5x + 3y = 80
2x + y = 30
55)
Solve the system of equations.
46)
7x + 7y + z = 1
x + 8y + 8z = 8
9x + y + 9z = 9
2x + 9y = -8
13
1
x + y = 3
3
Solve the problem. Round your answer to the nearest
tenth.
56) A wagon is pulled horizontally by exerting a
force of 60 pounds on the handle at an angle of
25° to the horizontal. How much work is done
in moving the wagon 50 feet?
57) Find the work done by a force of 200 pounds
acting in the direction -i + 2j in moving an
object 75 feet from (0, 0) to (-75, 0).
68) If v = 9i + 12j, find v .
69) If v = 4i + 5j, what is 9v ?
Graph the function.
x2 + x - 6
70) f(x) = x2 - x - 12
12
y
10
8
Decompose v into two vectors v1 and v2 , where v1 is
parallel to w and v2 is orthogonal to w.
58) v = -3i + 4j,
6
4
w = 3i + j
State whether the vectors are parallel, orthogonal, or
neither.
59) v = 3i + 4j,
w = 4i - 3j
2
-10 -8 -6 -4 -2
-2
2
4
6
8 10
-4
-6
-8
-10
Solve the problem.
60) Which of the following vectors is parallel to v
= -10i - 8j?
-12
71) f(x) = Find the angle between v and w. Round your answer to
one decimal place, if necessary.
61) v = -5i + 7j,
w = -6i - 4j
x2 + 7x + 12
x2 + 5
6
y
4
Find the dot product v · w.
62) v = -9i + 5j,
w = 15i - 9j
Solve the problem.
63) v = 15, α = 135°
2
-6
-4
-2
2
-2
-4
64) v = 11, α = 270°
65) The points (3, 6), (9, 2), (-3, -4) and (-9, 0) are
the vertices of a parallelogram ABCD. Find the
new vertices of a parallelogram AʹBʹCʹDʹ if it is
translated by the vector v = -8, 6 .
Find the unit vector having the same direction as v.
66) v = 12i - 5j
Find the indicated quantity.
67) If v = 3i - 5j and w = -7i + 4j, find 3v - 4w.
-6
4
6 x
x
72) f(x) = x - 2
2
x - x - 42
Find the domain of the rational function.
x
81) G(x) = 3
x - 216
y
6
5
82) Q(x) = 4
3
x + 5
2
x - 64x
2
1
-12 -10 -8 -6 -4 -2
-1
2
4
6
8 10 12 x
-2
Use the given zero to find the remaining zeros of the
function.
83) f(x) = x3 + 6x2 - 14x + 16; zero: 1 + i
-3
-4
-5
84) f(x) = x3 - 2x2 - 11x + 52; zero: -4
-6
Give the equation of the horizontal asymptote, if any, of
the function.
5x3 - 7x - 7
73) P(x) = 3x + 2
74) G(x) = 75) R(x) = x(x - 1)
x3 + 16x
-3x2
x2 + 2x - 99
Give the equation of the oblique asymptote, if any, of the
function.
9x2 - 4x - 8
76) H(x) = 7x2 - 9x + 6
x2 - 3x + 4
77) T(x) = x + 5
78) R(x) = x + 4
x2 - 49
Find the vertical asymptotes of the rational function.
x + 9
79) F(x) = x4 + 36
80) G(x) = x + 11
2
x + 49x
85) f(x) = x5 - 10x4 + 42x3 -124 x2 + 297x - 306 ;
zero: 3i
Find all zeros of the function and write the polynomial as
a product of linear factors.
86) f(x) = x3 - x2 + 4x - 4
87) f(x) = x3 + 6x2 + 17x + 18
88) f(x) = x4 + 5x3 + 10x2 + 20x + 24
Form a polynomial f(x) with real coefficients having the
given degree and zeros.
89) Degree: 4; zeros: -1, 2, and 1 - 2i.
90) Degree: 4; zeros: 2i and -3i
Information is given about a polynomial f(x) whose
coefficients are real numbers. Find the remaining zeros of
f.
91) Degree 5; zeros: 2, 5 + 5i, -2i
Use the Rational Zeros Theorem to find all the real zeros
of the polynomial function. Use the zeros to factor f over
the real numbers.
92) f(x) = 5x3 - 7x2 - 8x + 4
93) f(x) = 2x4 - 5x3 + 9x2 - 15x + 9
Find the intercepts of the function f(x).
94) f(x) = x3 + 2x2 - 9x - 18
95) f(x) = 5x4 - 20x3 + 21x2 - 4x + 4
96) f(x) = 4x5 (x + 5)3
97) f(x) = (x + 2)(x - 6)(x + 6)
98) f(x) = (x + 1)(x - 3)(x - 1)2
List the potential rational zeros of the polynomial
function. Do not find the zeros.
99) f(x) = -2x3 + 3x2 - 4x + 8
100) f(x) = 5x5 - 4x2 + 3x - 1
101) f(x) = x5 - 2x2 + 4x + 5
Answer Key
Testname: REVIEW 1
22) x = 11, y = -5; (11, -5)
23) inconsistent
24) y = -9x + 3, where x is any real number
or {(x, y) | y = -9x + 3, where x is any real number}
25) x = 0, y = 5, z = -4; (0, 5, -4)
26) inconsistent
27) x = -3z - 5, and y = z + 2, where z is any real number
or {(x, y, z) |x = -3z - 5, and y = z + 2, where z is any
real number}
28) x = 8, y = 5 or (8, 5)
29) x = 2, y = 0; x = -2, y = 0 or (2, 0), (-2, 0)
7
-6
+ 30)
x - 6 x - 7
1)
y
10
5
(9, 9)
(3, 3)
(9, 2.1)
(0, 1)
-10
(12, 2.3)
(3, 1)
-5
5
10
x
15
-5
-10
2) ± 1
7
, ± , ± 1, ± 7
11
11
3) -2, 2; f(x) = (x - 2)(x + 2)(x2 + 16)
4) 2 + 5i, -4i
5) f(x) = x3 + 6x2 - 14x + 16
6) 2i, 7, -7
7) {x|x ≠ -8, x ≠ 8}
8) x = -3, x = 3
9) y = x + 14
10) y = 1
11)
31)
5
6
-5
+ + x x - 3 (x - 3)2
32)
5
-5x + 4
+ x - 1 x2 + x + 1
33)
1
5x - 6
+ 2
x + 3 (x2 + 3)2
34)
3x
3x - 4
+ 2
x + 2 (x2 + 2)2
35)
5
-5x + 3
+ x - 1 x2 + x + 1
36)
2
4x - 6
+ x - 2 x2 + 5
37)
6
2
6
3
9
- - + - x x2 x + 1 (x + 1)2 (x + 1)3
38)
3
-2
+ x + 3 x + 2
39)
20
-6
-7
+ + x
x + 1 x - 1
40)
1
1
+ x - 6 x + 1
41)
4
7
-2
+ + x
x + 6 x - 2
y
5
-10
-5
5
10
x
-5
12) 12i + 6j
3
4
13) u = - i + j
5
5
14) v = 87
29
10 i + 10
10
10 j
10 j
15) 4
16) 37.9°
17) w = 2i - 2j
18) Orthogonal
19) v1 = 4i + 4j, v2 = -3i + 3j
20) 11.3 ft-lb
21) x = 12, y = -2; (12, -2)
or
v = - 87
10
10 i - 29
10
42) x = 2, y = 3; x = 2, y = -3; x = -2, y = 3; x = -2, y = -3
or (2, 3), (2, -3), (-2, 3), (-2, -3)
43) x = 3 3, y = 3 or (3 3, 3)
44) x = 2, y = 4 or (2, 4)
6
27
45) x = 0, y = 3; x = - , y = - 7
7
or 0, 3 , - 27
6
, - 7
7
46) x = 0, y = 0, z = 1; (0, 0, 1)
47) x = 1, y = -2, z = 3; (1, -2, 3)
48) x = 1, y = 2, z = 4; (1, 2, 4)
Answer Key
Testname: REVIEW 1
49) a = 2, b = -3, c = 2
50) a = 1, b = 6, c = -5
51) x = 12, y = -2; (12, -2)
52) x = -4, y = 0; (-4, 0)
1
1
53) x = 18, y = ; 18, 2
2
71)
12
y
8
4
54) x = 10, y = 10; (10, 10)
55) x = 5, y = -2; (5, -2)
56) 2718.9 ft-lb
57) 6708.2 ft-lb
1
3
9
3
58) v1 = - i - j, v2 = - i + j
2
2
2
2
-12
-8
-4
4
8
12 x
-4
-8
-12
59) Orthogonal
60) w = 20i + 16j
61) 88.2°
62) -180
15 2
15 2
63) v = - i + j
2
2
72)
y
6
5
4
3
64) v = -11j
65) (-5, 12), (1, 8), (-11, 2), (-17, 6)
5
12
66) u = i - j
13
13
2
1
-12 -10 -8 -6 -4 -2
-1
2
4
6
8 10 12 x
-2
67) 37i - 31j
68) 15
69) 9 41
70)
-3
-4
-5
-6
12
y
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
-12
2
4
6
8 10
x
73) none
74) y = 0
75) y = -3
76) none
77) y = x - 8
78) none
79) none
80) x = 0, x = -49
81) {x|x ≠ 6}
82) {x|x ≠ 0, x ≠ 64}
83) 1 - i, -8
84) 3 + 2i, 3 - 2i
85) -2, -3i, -4 - i, -4 + i
86) f(x) = (x - 1)(x + 2i)(x - 2i)
87) f(x) = (x + 2)(x + 2 + i 5)(x + 2 - i 5)
88) f(x) = (x + 3)(x + 2)(x - 2i)(x + 2i)
89) f(x) = x4 - 3x3 + 5x2 - x - 10
90) f(x) = x4 + 13x2 + 36
91) 5 - 5i, 2i
2
92) -1, , 2; f(x) = (5x - 2)(x - 2)(x + 1)
5
Answer Key
Testname: REVIEW 1
3
93) 1, ; f(x) = (x - 1)(2x - 3)(x2 + 3)
2
94) x-intercepts: -3, -2, 3; y-intercept: -18
95) x-intercept: 2; y-intercept: 4
96) x-intercepts: 0, -5; y-intercept: 0
97) x-intercepts: -2, -6, 6; y-intercept: -72
98) x-intercepts: -1, 1, 3; y-intercept: -3
1
99) ± , ± 1, ± 2, ± 4, ± 8
2
100) ± 1, ± 1
5
101) ± 1, ± 5