Test 3 Review- ch 3 and 4- 1314
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
1) Given f (x) = a(x - h)2 + k (a ≠ 0), the vertex of the parabola is the point
.
2)
3)
Given f (x) = a(x - h)2 + k, if a < 0, then the parabola opens
1)
.
The graph of f (x) = a(x - h)2 + k (a ≠ 0) is a parabola and the axis of symmetry is
.
the line given by x =
2)
3)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine whether the graph of the parabola opens upward or downward and determine the range.
4) f (x) = -2(x + 1)2 + 4
4)
A) Downward
B) Upward
Range: (-∞, 4]
Range: [-1, ∞)
C) Upward
D) Downward
Range: [4, ∞)
Range: (-∞, -1]
1
Sketch the function and determine the axis of symmetry.
5) f (x) = -(x + 2)2 + 9
A) Axis of symmetry: x = 9
B) Axis of symmetry: x = -9
10 y
C)
10 y
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
-2
2
4
6
8 10 x
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
Axis of symmetry: x = 2
D) Axis
10 y
2
4
6
8 10 x
6
8 10 x
of symmetry: x = -2
10 y
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
-2
5)
2
4
6
8 10 x
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
2
2
4
Identify the graph of the function.
6) h(x) = (x - 4)2 + 1
6)
A)
B)
y
y
-6
-4
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
C)
2
4
6
x
2
4
6
x
D)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
Identify the vertex, axis of symmetry, and intercepts for the graph of the function.
7) y = x2 + 8x + 11
A) Vertex at (-4, -5); axis: x = -4; x-intercepts: (-4 - 5, 0) and (-4 + 5, 0) ;
y-intercept: (0, 11)
B) Vertex at (4, 59); axis: y = 59; x-intercepts: (-4 - 5, 0) and (-4 + 5, 0) ;
y-intercept: (0, 11)
C) Vertex at (-4, -5); axis: y = -5; x-intercepts: none; y-intercept: (-4 - 5, 0)
D) Vertex at (4, 59); axis: x = 4; x-intercepts: none; y-intercept: (-4 + 5, 0)
3
7)
Write the function in vertex form and determine the range.
8) f (x) = -3x2 + 30x - 80
A) (x) = -3(x - 5)2 - 5
B) (x) = -3(x + 5)2 - 5
Range: (-∞, -5]
Range: [5, ∞)
2
C) (x) = -3(x + 5) - 5
D) (x) = -3(x - 5)2 - 5
Range: (-∞, -5]
Range: [5, ∞)
Determine the x- and y-intercepts for the given function.
9) f (x) = 2x2 + 4x - 6
A) x-intercepts: (-3, 0) and (1, 0)
B)
y-intercept: (0, -6)
C) x-intercept: none
D)
y-intercept: (0, -6)
9)
x-intercept: (-1, 0)
y-intercept: (0, -8)
x-intercept: (-1, 0)
y-intercept: (0, 8)
Solve the problem.
10) The daily profit in dollars made by an automobile manufacturer is
P(x) = -35x2 + 1540x + 315
where x is the number of cars produced per shift. Find the maximum possible daily
profit.
A) $15,875
B) $17,255
C) $14,667
D) $18,635
11) An
automobile manufacturer can produce up to 300 cars per day. The profit made from
the sale of these vehicles can be modeled by the function P(x) = -10x2 + 2000x - 36,000,
where P(x) is the profit in dollars and x is the number of automobiles made and sold.
How many cars should be made and sold to maximize profit?
A) 100
B) 50
C) 2000
D) 200
Determine the end behavior of the graph of the function.
12) f (x) = -5x5 + 9x4 - 2x3 + 2
A) Up left and up right
B) Down left and down right
C) Down left and up right
D) Up left and down right
Find the zeros of the function and state the multiplicities.
13) f (x) = -4x3 (x + 1)4 (x - 5)6
A) -1 (multiplicity 4), 5 (multiplicity 6)
B) 0 (multiplicity 3), 1 (multiplicity 4), -5 (multiplicity 6)
C) 0 (multiplicity 3), -1 (multiplicity 4), 5 (multiplicity 6)
D) 1 (multiplicity 4), -5 (multiplicity 6)
4
8)
10)
11)
12)
13)
A table of values is given for Y1 = f (x). Determine whether the intermediate value theorem guarantees
that the function has a zero on the given interval.
14) Y 1 = 14x4 - 12x3 + 175x2 - 477x + 249 [2, 3]
14)
X
-4
-3
-2
-1
0
1
2
3
Y1
9309
4713
2223
927
249
-51
123
1203
X=-4
A) No
B) Yes
5
Determine if the graph can represent a polynomial function. If so, assume the end behavior and all
turning points are represented on the graph.
15)
15)
5
y
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-2
-3
-4
-5
a. Determine the minimum degree of the polynomial based on the number of turning
points.
b. Determine whether the leading coefficient is positive or negative based on the end
behavior and whether the degree of the polynomial is odd or even.
c. Approximate the real zeros of the function, and determine if their multiplicity is odd
or even.
A) a. Minimum degree 3
b. Leading coefficient positive degree odd
c. -3, 0, and 2 (each with odd multiplicity)
B) a. Minimum degree 3
b. Leading coefficient negative degree odd
c. -3 (odd multiplicity), 0 (even multiplicity), 2 (even multiplicity)
C) a. Minimum degree 2
b. Leading coefficient negative degree even
c. -3 (odd multiplicity), 0 (even multiplicity), 2 (even multiplicity)
D) Not a polynomial function.
6
Sketch the function.
16) f (x) = -0.1(x + 4)2 (x + 1)3
16)
A)
B)
5 y
5 y
4
4
3
3
2
2
1
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5x
-5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
C)
1
2
3
4
5x
1
2
3
4
5x
D)
5 y
5 y
4
4
3
3
2
2
1
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5x
-5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
7
17)
f (x) = -x4 + 16x2 - 48
17)
A)
B)
y
y
5
5
-5
5
x
-5
-5
5
x
-5
C)
D)
60
50
40
30
20
10
-5
-10
-20
-30
-40
-50
-60
y
60
50
40
30
20
10
5 x
-5
y
-10
-20
-30
-40
-50
-60
5 x
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
18) The polynomial x2 - x - 12 factors as (x - 4)(x + 3). What is the quotient of
(x2 - x - 12) ÷ (x - 4)? What is the remainder?
19)
Given a polynomial f (x), the factor theorem indicates that if f (c) = 0, then x - c
is a
of f (x). Furthermore, if x - c is a factor of f (x), then f (c) =
18)
19)
.
20) The
remainder theorem indicates that if a polynomial f (x) is divided by x - c,
then the remainder is
.
20)
21) Answer
21)
true or false. If (x + 3) is a factor of a polynomial, then 3 is a zero of the
polynomial.
8
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use long division to divide.
22) (x5 - 3x4 - 21x2 - 21x - 36) ÷ (x2 + 4)
5x
A) x3 - 3x2 - 4x - 9 2
x +4
C) x3
- 7x2 + 7 -
22)
-49x + 36
2
x +4
B) x3
- 3x2 - 9x - 9
D) x3
- 3x2 + 4x - 33 -
5x + 168
x2 + 4
Solve the problem.
23) The following table represents the result of a synthetic division.
2
-4
9
8
-8
-4
-8
2
20
1
10
12
Use x as the variable. Identify the divisor.
A) x + 2
C) x - 2
B) -4x3
+ 9x2 + 8x - 8
D) 12
Use synthetic division to divide the polynomials.
24) (s2 + 5s + 4) ÷ (s + 1)
8
A) s - 4
B) s - 4 +
s+1
25)
23)
24)
C)
s+4
D) s -
x5 - 243
x-3
4-
8
s+1
25)
A) x4
+ 3x3 + 9x2 + 27x + 81
C) x4
- 3x3 + 9x2 - 27x + 81 -
486
x-3
B) x4
+ 81
D) x4
- 81
Use the remainder theorem to determine if the given binomial is a factor of f (x).
= x3 + 4x2 - 54x - 216;
A) No
26) f (x)
x +6 6
26)
B) Yes
Use the factor theorem to determine if the given binomial is a factor of f (x).
27) f (x) = x4 + 3x3 - 5x2 + 7x - 6; x - 2
A) Yes
B) No
9
27)
Solve the problem.
28) Use synthetic division and the remainder theorem to determine if [x - (3 + 2i)] is a factor
28)
2
of f (x) = x - 6x + 13.
A) No
B) Yes
Use the factor theorem to factor the polynomial.
29) Factor
f (x) = 10x3 - 37x2 + 40x - 12 given that
f (x) = (2x - 1)(5x - 6)(x - 2)
C) f (x) = (2x - 1)(5x + 6)(x + 2)
A)
1
is a zero.
2
B)
D)
29)
f (x) = (2x + 1)(5x - 6)(x + 2)
f (x) = (2x + 1)(5x + 6)(x - 2)
Write a polynomial f (x) that meets the given conditions. Answers may vary.
3
30) Degree 5 polynomial with zeros 1 and (each with multiplicity 1) and 0 (with
5
30)
multiplicity 3).
f (x) = 5x5 - 2x4 - 3x3
C) f (x) = 3x5 - 8x4 - 5x3
f (x) = 3x5 + 2x4 - 5x3
D) f (x) = 5x5 - 8x4 + 3x3
B)
A)
List the possible rational zeros.
31) f (x) = -49x4 + 5x3 + 7x + 14
2
2
1
1
A) ±1, ± , ±
, ±7, ±2, ±14, ± , ±
7 49
7 49
C)
31)
7
49
1
1
±1, ± , ± , ±7, ± , ±49, ±
7 49
2
2
Find all the zeros.
32) f (x) = x3 + 10x2 + 36x + 40
A) 2, -4 ± 2i
B) 2, 4 ± 2i
B) 1,
2 2
1 1
,
, 7, 2, ,
7 49
7 49
D) -1,
-
2
2
1
1
, - , -7, -2, - , 7 49
7 49
32)
C)
-2, -4 ± 2i
D) -2,
A polynomial f (x) and one of its zeros are given. Find all the zeros.
33) f (x) = x4 - 4x3 + 15x2 + 20x - 100;
2 - 4i is a zero
A) ±5, 2 ± 4i
B) ± 5, ±2 - 4i
C) ±5, ±2 - 4i
4 ± 2i
33)
D) ±
5, 2 ± 4i
Write a polynomial f (x) that meets the given conditions.
34) Degree
3 polynomial with integer coefficients with zeros -6i and
f (x) = 7x3 - 6x2 + 252x - 216
C) f (x) = 49x3 - 42x2 - 468x + 216
6
7
f (x) = 49x3 - 126x2 + 540x - 216
D) f (x) = 7x3 - 6x2 - 252x + 216
A)
B)
10
34)
35) Polynomial
of lowest degree with real coefficients and with zeros -4 + 3i (multiplicity
1) and 0 (multiplicity 3)
B) f (x) = x5 + 8x + 7
A) f (x) = x5 + 8x4 + 7x3
C) f (x) = x5 + 8x + 25
D) f (x) = x5 + 8x4 + 25x3
Determine the number of possible positive and negative real zeros for the given function.
36) f (x) = -8x7 - 2x4 + 4x3 + 5x2 + 7x + 5
A) Positive: 4 or 2; Negative: 1
B) Positive: 1; Negative: 4 or 2
C) Positive: 4; Negative: 1
D) Positive: 1; Negative: 4
Refer to the graph of the function and complete the statements.
37)
The domain is ________.
The range is ________.
35)
36)
37)
y
15
10
5
-15
-10
-5
5
10
15
x
-5
-10
-15
A) (-∞,
∞);
(-∞, 8) ∪ (8, ∞)
C) (-∞, 8) ∪ (8, ∞);
(-∞, ∞)
B) (-∞,
-2) ∪ (-2, ∞);
(-∞, 8) ∪ (8, ∞)
D) (-∞, 8) ∪ (8, ∞);
(-∞, -2) ∪ (-2, ∞)
A polynomial f (x) and one of its zeros are given. Find all the zeros.
38) f (x) = x4 - 10x3 + 31x2 + 100x - 410;
5 - 4i is a zero
A) ± 10, 5 ± 4i
B) ±10, 5 ± 4i
C) ±10, ±5 - 4i
Determine the vertical asymptote(s) of the graph of the function.
x
39) f (x) =
2
x + 16
A) x =
4 and x = -4
C) x = 4 and x = 0
B) x =
4
D) None
11
38)
D) ±
10, ±5 - 4i
39)
a. Identify the horizontal asymptote (if any).
b. If the graph of the function has a horizontal asymptote, determine the point where the graph crosses the
horizontal asymptote.
7x2 - 6x + 3
40) f (x) =
40)
x2 + 4
y=7
25
b. - , 7
6
B) a.
y=0
b. Graph does not cross y = 0.
a. y = 0
3
b. , 0
4
D) a.
A) a.
C)
41) s(x)
No horizontal asymptote
b. Not applicable
2x - 10
= 2
x + 7x - 4
41)
y=0
5
b. (- , 0)
2
B) a.
y=0
b. (5, 0)
a. y = 5
b. (0, 5)
D) a.
A) a.
C)
No horizontal asymptote
b. Not applicable
Identify the asymptotes.
x3 - 2x2 - 6x + 1
42) f (x) =
x2 - 2
42)
A) Vertical
asymptotes: x = 2 and x = - 2
Horizontal asymptote: y = 1
B) Vertical asymptotes: x = 2 and x = - 2
Slant asymptote: y = -4x - 3
C) Vertical asymptotes: x = 2 and x = -2
Horizontal asymptote: y = 1
D) Vertical asymptotes: x = 2 and x = - 2
Slant asymptote: y = x - 2
12
Graph the function by using transformations of the graph of y =
43)
f (x) =
1
x2
.
1
-2
(x + 4)2
43)
A)
B)
10 y
10 y
5
-10
-5
5
5
x
-10
-5
-5
-5
-10
-10
C)
5
x
5
x
D)
10 y
10 y
5
-10
-5
5
5
x
-10
-5
-5
-5
-10
-10
13
Graph the function.
3x
44) f (x) =
2
x - 2x - 3
44)
A)
B)
y
y
5
5
-5
5
x
-5
-5
5
x
5
x
-5
C)
D)
y
y
5
-5
5
5
x
-5
-5
-5
14
45) g(x)
=
x3 + 3x2 - x - 3
45)
x2 + 2x
A)
B)
10 y
-10 -8 -6 -4 -2
10 y
8
8
6
6
4
4
2
2
-2
2
4
6
8
x
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
C)
2
4
6
8
x
2
4
6
8
x
D)
10 y
-10 -8 -6 -4 -2
10 y
8
8
6
6
4
4
2
2
-2
2
4
6
8
x
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
Solve the problem.
46) A power company burns coal to generate electricity. The cost C(x) (in $1000) to remove
x% of the air pollutants is given by
600x
C (x) =
.
100 - x
If the power company budgets $1.4 million for pollution control, what percentage of the
air pollutants can be removed? Round to two decimal places, if necessary.
A) 70%
B) 0.7%
C)
15
0.23%
D) 23.28%
46)
Solve the inequality. Write the solution set in interval notation.
47) (5x - 4)(7x - 8) < 0
4
8
4 8
A) -∞,
B)
∪ ,∞
,
5
7
5 7
C)
-∞,
4
8
∪ ,∞
5
7
D)
47)
4 8
,
5 7
48) d2
> 4d
A) (0, 4)
C) (-∞, 4]
49) (k
50)
0) ∪ (4, ∞)
D) (4, ∞)
49)
B) (-∞,
-8) ∪ (4, 5)
D) (-5, -4) ∪ (8, ∞)
x-1
≤0
x-5
10 - 5x
x2
A) (2,
52)
B) (-∞,
- 5)(k - 4)(k + 8) < 0
A) (-∞, -5) ∪ (-4, 8)
C) (-8, 4) ∪ (5, ∞)
A) (-5,
51)
48)
x2
x2 + 49
50)
-1)
B) (1,
5)
C)
D) (-5,
-1]
≤0
∞)
51)
B) (-∞,
2)
C)
(-∞, 2]
D) [2,
∞)
≥0
52)
A) (-∞,
0) ∪ (0, ∞)
C) (-∞, 7] ∪ [7, ∞)
53)
[1, 5)
B) (-∞,
∞)
D) {0}
3x
≥3
x-5
A) (5,
∞)
53)
B) [5,
∞)
C)
[0, ∞)
D) (0,
∞)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
54) Given the function f : = {(1, 2), (2, 3), (3, 4)} write the set of ordered pairs
representing f -1
55) A
function f is a
f, if a ≠ b, then f (a) ≠ f (b).
-
function if for a and b in the domain of
16
54)
55)
function defined by f (x) = x2 - 9 (is/is not) a one-to-one function, whereas
g(x) = x2 - 9; x ≥ 0 (is/is not) a one-to-one function.
56) The
56)
57) The
57)
58) A
58)
notation
is often used to represent the inverse of a function f and
not the reciprocal of f.
function defined by y = f (x) (is/is not) a one-to-one function if no horizontal
line intersects the graph of f in more than one point.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
A relation in x and y is given. Determine if the relation defines y as a one-to-one function of x.
59)
59)
x
y
3.0 7.45
-8.4 -8.07
2.4 -9.16
-1.5 7.45
A) No
B) Yes
Determine if the relation defines y as a one-to-one function of x.
60)
60)
5 y
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5x
-2
-3
-4
-5
A) No
B) Yes
Determine whether the two functions are inverses.
2
2 + 8x
61) f (x) =
and g (x) =
x-8
x
61)
A) Yes
B) No
17
A one-to-one function is given. Write an expression for the inverse function.
9-x
62) f (x) =
5
A) f -1(x)
=
9 - 5x
x
B) f -1(x)
=
C) f -1(x)
= 9 - 5x
D) f -1(x)
= 9 + 5x
B) f -1(x)
D) -1
3
63) f (x)
=
3
62)
9 + 5x
x
x-8
A) f -1(x)
C) -1
63)
3
=x -8
3
=x +8
f (x) = (x + 8)3
f (x) = (x - 8)
Given f (x), write an equation for f -1(x) and then graph f (x) and f -1(x) on the same coordinate system.
= x2 - 7; x ≤ 0
A) f -1(x) = x + 7
64) f (x)
64)
B) f -1(x)
=- x+7
10 y
10 y
10 x
-10
-10
C) f -1(x)
-10
D) f -1(x)
=- x+7
10 y
= x+7
10 y
10 x
-10
10 x
-10
10 x
-10
-10
-10
18
Graph the function and write the domain and range in interval notation.
65) f (x) = 7x
A)
65)
B)
y
y
10
10
-5
5
x
-5
Domain: (-∞, ∞)
Range: (0, ∞)
5
x
5
x
Domain: (0, ∞)
Range: (-∞, ∞)
C)
D)
y
y
10
-5
10
5
x
-5
Domain: (0, ∞)
Range: (-∞, ∞)
Domain: (-∞, ∞)
Range: (0, ∞)
19
Solve the problem.
66) Use the graph of y = 2x to graph the function. Write the domain and range in interval
notation.
f (x) = 2x + 1 - 4
A)
B)
10 y
10 y
5
-10
-5
5
5
x
-10
-5
-5
-5
-10
-10
Domain: (-∞, ∞)
Range: (1, ∞)
5
x
5
x
Domain: (-∞, ∞)
Range: (-1, ∞)
C)
D)
10 y
10 y
5
-10
-5
5
5
x
-10
-5
-5
-5
-10
-10
Domain: (-∞, ∞)
Range: (4, ∞)
67) Use
66)
Domain: (-∞, ∞)
Range: (-4, ∞)
1x
the graph of y =
to graph the function. Write the domain and range in interval
2
notation.
f (x) =
1 x+4
-3
2
20
67)
A)
B)
10 y
10 y
5
-10
-5
5
5
x
-10
-5
-5
-5
-10
-10
Domain: (-∞, ∞)
Range: (-∞, 3)
5
x
5
x
Domain: (-∞, ∞)
Range: (-3, ∞)
C)
D)
10 y
10 y
5
-10
-5
5
5
x
-10
-5
-5
-5
-10
-10
Domain: (-∞, ∞)
Range: (3, ∞)
Domain: (-∞, ∞)
Range: (-∞, -3)
21
68) Newton's
law of cooling indicates that the temperature of a warm object will decrease
exponentially with time and will approach the temperature of the surrounding air. The
temperature T(t) is modeled by T(t) = Ta + (T0 - Ta )e-kt. In this model, Ta represents the
temperature of the surrounding air, T0 represents the initial temperature of the object and
t is the time after the object starts cooling. The value of k is the cooling rate and is a
constant related to the physical properties of the object.
68)
A cake comes out of the oven at 380°F and is placed on a cooling rack in a 65°F kitchen.
After checking the temperature several minutes later, it is determined that the cooling
rate k is 0.042. Write a function that models the temperature T(t) (in °F) of the cake t
minutes after being removed from the oven.
= 65 + 315e-0.042t
C) T(t) = 65 + 315e0.042t
= 380 + 65e0.042t
D) T(t) = 65 + 380e0.042t
A) T(t)
B) T(t)
69) Scientists
often use a process called carbon dating to estimate the age of archaeological
finds. The process measures the amount of carbon-14, a radioactive isotope with a
half-life of 5,730 years. If a sample of wood from an ancient artifact had 5 grams of
carbon-14 initially, the amount remaining, in grams, is given by
1 t/5,730
A(t) = 5
2
where t is the number of years since the tree died. How many grams would be present
after 5,730 years?
A) 1.25
g
B) 0.27
g
C)
2.5 g
D) 10
g
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
70) Given positive real numbers x and b such that b ≠ 1, y = logb x is the
70)
function base b and is equivalent to b y = x.
71) The
inverse of an exponential function base b is the
72) Given y
= log x, the base is understood to be
is understood to be
.
73)
function base b.
. Given y = ln x, the base
f (x) = logb x and g(x) = bx are inverse functions. Therefore, logb bx =
log x
b
and b
=
.
22
71)
72)
73)
69)
Write the equation in exponential form.
1
74) log3
= -5
243
74)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the equation in logarithmic form.
75) 34 = 81
A) log 81 = 3
B) log 3 = 4
4
81
75)
C)
log81 4 = 3
D) log
3
81 = 4
Simplify the expression.
1
76) ln
e6
A) -
1
6
76)
B)
1
6
C)
-6
D) 6
77) log 0.0001
A)
1
4
77)
B) -4
C)
1
3
D) -3
5
log (x + 4)
78) 6 6
A) 4
78)
C) x5
B) 5
23
D) x5
+4
Graph the function.
79) y = log x
8
79)
A)
B)
-8
-6
-4
-2
8 y
8 y
6
6
4
4
2
2
-2
2
4
6
8 x
-8
-6
-4
-2
-2
-4
-4
-6
-6
-8
-8
C)
2
4
6
8 x
2
4
6
8 x
D)
-8
-6
-4
-2
8 y
8 y
6
6
4
4
2
2
-2
2
4
6
8 x
-8
-6
-4
-2
-2
-4
-4
-6
-6
-8
-8
Simplify the expression.
6
80) ln ex - 2
A) 6
80)
B) x6
C) x6
24
-2
D) -2
a. Use transformations to graph the function.
b. Write the domain and range in interval notation.
c. Determine the vertical asymptote.
81) log y = (x - 4)
3
81)
A) a.
-8
B) a.
-6
-4
-2
8 y
8 y
6
6
4
4
2
2
-2
2
4
6
8 x
-8
-6
-4
-2
-4
-2
-2
-4
-4
-6
-6
-8
-8
b. domain: (0, ∞), range (-∞, ∞)
c. vertical asymptote: x = 0
C) a.
-8
-6
8 y
6
6
4
4
2
2
2
4
6
4
6
8 x
b. domain: (4, ∞), range (-∞, ∞)
c. vertical asymptote: x = 4
D) a.
8 y
-2
2
8 x
-8
-6
-4
-2
-2
2
4
6
-4
-4
-6
-6
-8
-8
b. domain: (0, ∞), range (-∞, ∞)
c. vertical asymptote: x = 0
b. domain: (-4, ∞), range (-∞, ∞)
c. vertical asymptote: x = -4
25
8 x
Solve the problem.
82) Use transformations of the graph of y = log x to graph the function.
82)
5
y = log5(x - 4) - 3
A)
B)
10 y
10 y
5
-10
-5
5
5
x
-10
-5
-5
-5
-10
-10
C)
5
x
5
x
D)
10 y
10 y
5
-10
-5
5
5
x
-10
-5
-5
-5
-10
-10
Write the domain in interval notation.
83) log (4 - x)2
7
83)
A) (-∞,
-4) ∪ (4, ∞)
C) (-∞, ∞)
B) (-4,
4)
D) (-∞, 4) ∪ (4, ∞)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
84) The product property of logarithms indicates that logb (xy) =
positive real numbers b, x, and y where b ≠ 1.
26
for
84)
85) The
change-of-base formula indicates that logb x can be written as a ratio of
□
logarithms with base a as: logb x = .
□
85)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the product property of logarithms to write the logarithm as a sum of logarithms. Then simplify if
possible.
86) log 64x
86)
8
A) 2
C)
+ log8 x
B) 2
log8 x + log8 64
log8 x
D) (log
8
64)(log8 x)
87) log (xy)
13
87)
A) log
13
C)
x + log13 y
B) log
13
log13 x × log13 y
x - log13 y
D) (log x)
13
·y
Use the quotient property of logarithms to write the logarithm as a difference of logarithms. Then
simplify if possible.
t
88) log
88)
6 36
A) log
6
C)
t-2
B)
log6 t - log6 36
D)
Apply the power property of logarithms.
89) log(3t - 7)4
A) 4 log 3t - 4 log 7
C) 4 log 3 + 4 log t - 4 log 7
90) ln
log6 t
2
log6 t
log6 36
89)
B) 4
log(3t - 7)
D) 4 log 3 · 4 log t - 4 log 7
5 9
x
A) 9ln x - 5ln x
90)
B)
9ln x
5ln x
C)
1
ln x · 9ln x
5
91) ln 11kt
D)
9
ln x
5
91)
A) 11
B) kt
C)
ln k · 11 ln t
kt ln 11
D) 11
27
+ ln 11
ln kt
Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible.
92) log x4 y7
92)
8
A) log x4 +
8
C)
log8 y7
B) 11(log
8
4log8 x + 7log8 y
D) 28log
8
x + log8 y)
x + 28log8 y
Write as the sum or difference of logarithms and fully simplify, if possible. Assume the variable
represents a positive real number.
e4
5
93) ln
93)
x6 + y
A)
4 1
+ ln(x6 + y)
5 5
B)
4
1
ln e - ln(x6 + y)
5
5
C)
4 6
1
- ln x + ln y
5
5 5
D)
4 1
- ln(x6 + y)
5 5
Write the logarithmic expression as a single logarithm with coefficient 1, and simplify as much as
possible.
94) log 168 - log 4 - log 7
94)
6
6
6
A) 157
95)
B) 1
C)
log6 179
D) log
6
4704
1
[log (a2 + 9a) - log (a + 9)]
5
A) log
C)
96) 8
log
log x A) log
95)
5 2
5
1/5
a +a-9
B) log(a3
+ 9a2 + 81a)
a
D) log
1/5
(a2 + a - 9)
5
1
log y - log z
6
6
x8
6
yz5
96)
B)
x8
6
C)
yz5
log (x8 y
6 5
z )
6
D) log(x8
Solve the problem.
97) Given that log 3 ≈ 0.4771 and log 8 ≈ 0.9031, use the properties of logarithms to
approximate the following. Do not use a calculator.
3
log
64
A) 2.2833
B) -1.3291
C)
28
-0.3385
D) 0.5850
yz5 )
97)
Solve the equation.
98) 32x + 3 = 43x + 7
A) {1}
99) 32x+4
98)
B) {-8}
C)
{-2}
D) {
}
= 26x
A) No
solution
99)
B)
9
5
C) -
32
5
D) {20}
Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4
decimal places if necessary.
100) log (x2 - 2x) = log 35
100)
A) {7}
B) {-7, 5}
C) {7, -5}
D) { }
Use the change-of-base and a calculator to approximate the logarithm to 4 decimal places.
101) log 13
4
A) 0.5405
B) 0.6412
C)
1.8502
D) 0.1066
Determine if the given value of x is a solution to the logarithmic equation.
102) log (x - 8) = 2 - log x; x = 9
3
3
A) Yes
102)
B) No
Solve the equation.
103) log r + log (r - 6) = 4
2
2
A) ∅
103)
B) {8}
C)
{2}
D) {11}
Solve the logarithmic equation.
104) log x = 1 - log(x + 3)
A) {2}
104)
B) {2,
-5}
C)
3
,5
2
D) {1,
-3}
Solve the problem.
105) If $25,000 is invested in an account earning 4.3% interest compounded continuously,
determine how long it will take the money to triple. Round to the nearest year. Use the
model A = Pert where A represents the future value of P dollars invested at an interest
rate r compounded continuously for t years.
A) 255
101)
years
B) 32
years
C)
29
26 years
D) 3
years
105)
Find the vertex of the parabola.
106) h(x) = -4x2 + 24x - 41
A) (-5, -3)
B) (-3, -5)
106)
C)
30
(0, -41)
D) (3,
-5)
Answer Key
Testname: TEST3 REVIEW
(h, k)
2) downward
3) h
1)
4) A
5) D
6) A
7) A
8) A
9) A
10) B
11) A
12) D
13) C
14) A
15) A
16) D
17) C
18) Quotient:
19) factor;
20)
(x + 3); Remainder: 0
0
f (c)
21) False
22) A
23) C
24) C
25) A
26) A
27) B
28) B
29) A
30) A
31) A
32) C
33) D
34) A
35) D
36) B
37) D
38) A
39) D
40) A
41) B
42) D
43) C
44) C
45) D
46) A
47) D
48) B
31
Answer Key
Testname: TEST3 REVIEW
49) B
50) C
51) D
52) B
53) A
54) {(2,
1), (3, 2), (4, 3)}
55) one-to-one
56) is not; is
57)
f
-1
58) is
59) A
60) A
61) A
62) C
63) B
64) C
65) A
66) D
67) B
68) A
69) C
70) logarithmic
71) logarithmic
72) 10; e
73) x ; x
74) 3-5 =
1
243
75) D
76) C
77) B
78) D
79) D
80) C
81) B
82) C
83) D
84) logb
85)
x + logb y
loga x
loga b
86) A
87) A
88) A
89) B
32
Answer Key
Testname: TEST3 REVIEW
90) D
91) C
92) C
93) D
94) B
95) C
96) A
97) B
98) A
99) D
100) C
101) C
102) A
103) B
104) A
105) C
106) D
33