Pre-Calculus
Mr. Kellogg
Semester 1 Final Exam
Please Do Not Write on this test!
6)
MULTIPLE CHOICE. Choose the one alternative that
best completes the statement or answers the question.
y
10
Solve the equation algebraically.
1) v2 + 5 = 7 - 2v2
A) ±
3
2
C) ±
2
3
5
B) ±
5
7
D) ±
2
5
-10
-5
5
10 x
-5
-10
2) x - 10x - 25 = 0
A) - 5
C) 25 ; -1
B) - 25 ; 1
D) 5
A) No
Find the domain of the given function.
5
7) f x = x2
Determine whether the equation defines y as a function
of x.
3) x = y 2 + 8
A) No
A) All real numbers
B) [0,∞)
C) (-∞,0) ∪ (0,∞)
D) (-∞,3] ∪ [3,∞)
B) Yes
4) y = 3x2 + 7x - 4
A) Yes
B) No
Determine algebraically whether the function is even,
odd, or neither even nor odd.
8) f(x) = 2x2 - 1
Determine whether the graph is the graph of a function.
5)
A) Neither
B) Even
C) Odd
y
9) f(x) = x2 + 10
A) Even
B) Neither
C) Odd
x
A) No
B) Yes
Find the asymptote(s) of the given function.
(x - 7)(x + 9)
10) h(x) = vertical asymptotes(s)
x2 - 4
B) Yes
A) x = 7, x = -9
C) x = -7, x = 9
11) f(x) = x - 6
vertical asymptotes(s)
x2 + 8x
A) x = 6
C) x = 8
1
B) x = 2, x = -2
D) None
B) x = -8
D) x = 0, x = -8
A)
Describe how the graph of y=x 2 can be transformed to the
graph of the given equation.
12) y = (x + 14)2 + 7
y
10
A) Shift the graph of y = x2 left 14 units and
then up 7 units.
B) Shift the graph of y = x2 right 14 units
5
and then up 7 units.
-10
C) Shift the graph of y = x2 down 14 units
and then right 7 units.
D) Shift the graph of y = x2 up 14 units and
-5
5
10
x
5
10
x
5
10
x
5
10
x
-5
then left 7 units.
-10
B)
Describe how to transform the graph of f into the graph of
g.
13) f(x) = (x + 9)2 and g(x) = -(x - 4)2
y
10
A) Shift the graph of f left 13 units and
reflect across the x-axis.
B) Shift the graph of f right 13 units.
C) Shift the graph of f down 13 units and
reflect across the y-axis.
D) Shift the graph of f right 13 units and
reflect across the x-axis
5
-10
-5
-5
-10
Sketch the graph of y 1 as a solid line or curve. Then
sketch the graph of y 2 as a dashed line or curve by one or
C)
more of these: a vertical and/or horizontal shift of the
graph y1 , a vertical stretch or shrink of the graph of y 1 , or
y
10
a reflection of the graph of y 1 across an axis.
14) y1 = ∣x∣; y2 = ∣x - 2∣
5
y
10
-10
-5
-5
5
-10
-10
-5
5
10
x
D)
-5
y
10
-10
5
-10
-5
-5
-10
2
1
15) y1 = ∣x∣, y2 = ∣x + 6∣ - 2
4
D)
y
10
y
10
5
5
-10
-10
-5
5
10
-5
5
x
10
x
10
x
-5
-5
-10
-10
A)
Graph the piecewise-defined function.
16)
6x + 2, if x < 0
y
10
y(x) =
2x2 - 3, if x ≥ 0
5
y
-10
-5
5
10
x
10
-5
5
-10
-10
B)
-5
5
10
x
-5
y
10
-10
5
A)
y
-10
-5
5
10
x
10
-5
-10
-10
C)
y
10
-10
5
-10
-5
5
10
x
-5
-10
3
17) f(x) = 3(x + 2)2 - 5
A)
B)
y
10
y
10
5
-10
-5
5
10
x
-10
-5
10
x
10
x
10
x
10
x
-10
-10
C)
B)
y
10
y
10
5
-10
-5
5
10
x
-10
-5
-10
-10
D)
C)
y
10
y
10
5
-10
-5
5
10
x
-10
-5
-10
-10
D)
Match the equation to the correct graph.
y
10
-10
-10
4
25) f x = x2 - 2x + 7; d x = x - 3 (Write answer in
fractional form)
f(x)
10
A) = x + 1 + x - 3
x - 3
Write the quadratic function in vertex form.
18) y = x2 + 2x + 7
A) y = (x - 1)2 + 6
C) y = (x + 1)2 + 6
B) y = (x - 1)2 - 6
D) y = (x + 1)2 - 6
Describe the end behavior of the polynomial function by
finding lim f x and lim f x .
x→∞
x→-∞
19) f x = -2x2 + 3x3 + 4x + 6
A) -∞, ∞
C) ∞, -∞
B) -∞, -∞
D) ∞, ∞
B)
f(x)
10
= x - 3 + x - 3
x - 3
C)
f(x)
3
= x - 3 + x - 3
x - 3
D)
f(x)
3
= x + 1 + x - 3
x - 3
Divide using synthetic division, and write a summary
statement in fraction form.
2x5 - x4 + 3x2 - x + 5
26)
x - 1
20) f x = (x - 3) ( 1 - x) (3x - 5)
A) ∞, -∞
B) ∞, ∞
D) -∞, -∞
C) -∞, ∞
A) 2x4 + x3 + 4x2 + 3x + Find the zeros of the function.
21) f x = x2 - 6x + 8
A) 4 and 2
C) -4 and -2
22) f x = x3 + 6x2 - 27x - 140
A) -7, 4, and -5
C) 7, -4, and 5
B) 2x4 - 3x3 + x + B) -4 and 2
D) 4 and -2
8
x + 1
6
x + 1
C) 2x4 + x3 + x2 + 4x + 3 + 8
x + 1
D) 2x4 + x3 - x2 + 2x + 1 + 6
x + 1
B) -7, -4, and 5
D) 7, 4, and -5
27)
Find a cubic function with the given zeros.
23) 7, -2, 6
A) f x = x3 - 11x2 + 16x + 84
3x5 + 4x4 + 2x2 - 1
x + 2
A) 3x4 - 2x3 + 4x2 + 6 + -13
x + 2
B) f x = x3 - 11x2 - 16x + 84
C) f x = x3 - 11x2 + 16x - 84
B) 3x4 - 2x3 + 6x2 - 12 + 23
x + 2
D) f x = x3 + 11x2 + 16 + 84
C) 3x4 + 2x3 + 4x2 + 8x + -15
x + 2
Divide f(x) by d(x), and write a summary statement in the
form indicated.
24) f x = x3 + 6x2 + 9x - 5; d x = x + 5 (Write
D) 3x4 - 2x3 + 4x2 - 6x + 12 + -25
x + 2
Find the remainder when f(x) is divided by (x - k)
28) f(x) = 3x4 + 7x3 + 5x2 - 6x + 41; k = -2
answer in polynomial form)
A) f x = x + 5 x2 + x + 4 - 25
B) f x = x + 5 x2 - x + 4 - 25
C) f x = x + 5 x2 + x + 4 + 25
D) f x = x + 5 x2 + x - 4 + 25
A) 13
B) 31
C) 14
D) 65
Write the polynomial in standard form and identify the
zeros of the function.
29) f(x) = (x - 5i)(x +5i)
A) f(x) = x2 + 25; zeros ± 5i
B) f(x) = x2 + 25; zeros ± 5
C) f(x) = x2 + 5ix + 25; zeros ± 5
D) f(x) = x2 - 25; zeros ± 5i
5
30) f(x) = (x + 2)(x + 2)(x + 3i)(x - 3i)
A) f(x) = x4 + 13x2 + 36; zeros 2 (mult. 2), ±
36)
3
4
32
8x
- = x - 8 x
x2 - 8x
A) x = B) f(x) = x4 + 4x3 - 5x 2 - 36x - 36; zeros -2
(mult. 2), ± 3
C) f(x) = x4 + 4x3 + 13x2 + 36x + 36; zeros -2
1
2
1
1
C) x = or - 2
2
1
1
B) x = or - 4
4
D) x = 2
(mult. 2), ± 3i
D) f(x) = x4 + 4x3 + 13x2 + 36x + 36; zeros 2
(mult. 2), ± 3i
Find the center, vertices, and foci of the ellipse with the
given equation.
x2 y2
+ = 1
37)
25
9
Write a polynomial function of minimum degree with
real coefficients whose zeros include those listed. Write
the polynomial in standard form.
31) 2i and 2
A) f(x) = x4 + 4x2 + 8
A) Center: (0, 0); Vertices: (-5, 0), (5, 0);
Foci: (0, -3), (0, 3)
B) Center: (0, 0); Vertices: (0, -5), (0, 5);
Foci: (-3, 0), (3, 0)
C) Center: (0, 0); Vertices: (-5, 0), (5, 0);
Foci: (-4, 0), (4, 0)
D) Center: (0, 0); Vertices: (0, -5), (0, 5);
Foci: (0, -4), (0, 4)
B) f(x) = x4 - 4x2 + 8
C) f(x) = x4 + 2x2 - 8
D) f(x) = x4 - 2x2 - 8
32) -3 and 3 + 2i
A) f x = x3 - 3x2 - 5x + 39
38)
B) f x = x3 - 3x2 - 5x - 39
C) f x = x3 - 3x2 - 5x + 40
A) Center: (1, 3); Vertices: (1, -17), (1, 23);
Foci: (1, -13), (1, 19)
B) Center: (1, 3); Vertices: (1, -17), (1, 23);
Foci: (-9, 3), (15, 3)
C) Center: (1, 3); Vertices: (-17, 3), (23, 3);
Foci: (-13, 1), (19, 1)
D) Center: (1, 3); Vertices: (-17, 3), (23, 3);
Foci: (1, -9), (1, 15)
D) f x = x3 + 3x2 - 5x + 39
For the given function, find all asymptotes of the type
indicated (if there are any)
x - 1
33) f(x) = , vertical
x2 - 4
A) x = 2, x = -2
C) x = 2
34) f(x) = B) x = -2
D) x = 1
Compute the exact value of the function for the given
x-value without using a calculator.
39) f(x) = - 3 · 13x for x = 1/3
x + 9
, horizontal
2
x + 9x + 3
A) y = 9
C) y = 0
B) None
D) y = x
Solve the equation.
6
35) x + 5 = x
A) x = -1 or x = 6
C) x = -5
(x - 1)2 (y - 3)2
+ = 1
144
400
-3
3
13
3
A) - 13 3
B)
3
C) - 3 13
D) - 13
Decide whether the function is an exponential growth or
exponential decay function and find the constant
percentage rate of growth or decay.
40) f(x) = 20,032 · 0.828 x
B) x = ± 6
D) x = -6 or x = 1
A) Exponential decay function; 0.172%
B) Exponential decay function; -17.2%
C) Exponential growth function; -17.2%
D) Exponential growth function; 0.172%
6
47) 4 log6 (5x + 6) + 2 log6 (3x + 8)
Evaluate the logarithm.
1
41) ln e3
A) log6 (5x + 6)4
(3x + 8)2
A) -6
B) -3
B) 8 log6 (5x + 6)(3x + 8)
3
C) - 2
3
D)
2
C) log6 (5x + 6)4 (3x + 8)2
D) log6 ((5x + 6)4 + (3x + 8)2 )
Write the expression using only the indicated logarithms.
48) log9 (x + y) using common logarithms
42) log3 9
A) 9
B) 3
C) 2
D) 6
A) log (x + y) log 9
log 9
B)
log (x + y)
Solve the equation by changing it to exponential form.
43) log x = 3
8
3
A) x = 8 3
B) x = log8 3
C) x = 3 8
C) log (x + y) + log 9
log (x + y)
D)
log 9
D) x = 8·3
Find the exact solution to the equation.
49) log4 (x - 1) = - 1
A) x = -0.75
C) x = 3
Rewrite the expression as a sum or difference or multiple
of logarithms.
17 r
44) log10 s
A) log10 (17 r) - log10 s
1
B) log10 s - log10 17 - log10 r
2
1
C) log10 17 + log10 r - log10 s
2
1
D) log10 17 · log10 r ÷ log10 s
2
45) ln x5 y4
A) 5ln x + 4ln y
B) ln x5 · ln y4
C) ln (5x)+ ln (4y)
D) ln (5x) · ln (4y)
Use the product, quotient, and power rules of logarithms
to rewrite the expression as a single logarithm. Assume
that all variables represent positive real numbers.
1
46) ln x
4
A) ln C) ln 4
x
4
x
B) ln x4
D) ln x - ln 4
7
B) x =5
D) x = 1.25
Answer Key
Testname: PRECALC_SEM1_FINALREVIEW
1) C
2) D
3) A
4) A
5) A
6) B
7) C
8) B
9) A
10) B
11) D
12) A
13) D
14) C
15) B
16) C
17) A
18) C
19) C
20) C
21) A
22) B
23) A
24) A
25) A
26) C
27) D
28) D
29) A
30) C
31) C
32) A
33) A
34) C
35) D
36) A
37) D
38) A
39) C
40) B
41) C
42) C
43) A
44) C
45) A
46) A
47) C
48) D
49) D
8