College Algebra ~ Review for Test 2
Sections 2.2 - 3.3
Find a point-slope form for the equation of the line
satisfying the conditions.
1) a) Slope -3, passing through (7, 4)
b) Passing through (-2, -8) and (-5, 2)
Write an equation in slope-intercept form for the line
shown.
2)
6
Graph the linear equation.
3
10) y = (x + 2) - 1
5
Identify the equation as either linear or nonlinear by
trying to write it in the form ax + b = 0.
11) a) 11 + 146x = 124x
b) x2 + 2x = x3 + 3
y
c) 7x2 - 3x = 7(x2 + 5) - 7x
4
Solve the equation symbolically.
12) 32x - 6 = 14
2
-6
-4
-2
2
4
6x
13) 9x - (2x - 1) = 2
-2
-4
14) 29t - 3 = 8t + 12
-6
15)
Write the slope-intercept form of the equation for the line
passing through the given pair of points.
3) (-9, 6) and (-4, 8)
4) (-6, 0) and (-9, -4)
Determine the equation of the line described. Put the
answer in the slope-intercept form.
5) Through (-2, 2), parallel to 4x - 7y = 6
6) Through (-8, -2), perpendicular to 8x + 7y =
-78
Find an equation of the line satisfying the following
conditions.
7) Vertical, passing through (-7, -9)
8) Horizontal, passing through (3, 5)
Solve.
9) The time T necessary to make an enlargement
of a photo negative is directly proportional to
the area A of the enlargement. If 135 seconds
are required to make a 15 in2 enlargement, find
the time required for a 70 in2 enlargement.
9x + 1 3x - 2 9
+
=
2
3
4
16) 9x2 + 3x - 4 = x(3 + 9x) - 4
Classify the equation as a contradiction, an identity, or a
conditional equation.
17) a) 15k - 93 = 3(5k - 33)
b) 4(5f + 1) = 20f + 4
c) 2(18t + 12) = 6(2t - 24)
Solve the problem.
18) Brand A soup contains 833 milligrams of
sodium. Find a linear function f that computes
the number of milligrams of sodium in x cans
of Brand A soup.
Solve the problem. Round your answer to the nearest
whole number.
19) A tree casts a shadow 15 m long. At the same
time, the shadow cast by a 48-cm tall statue is
92 cm long. Find the height of the tree.
Solve the problem.
20) A rectangular Persian carpet has a perimeter of
172 inches. The length of the carpet is 26 in.
more than the width. What are the dimensions
of the carpet?
Math 1314, Review for Test 2
Page 2
Use the given graphs of y = ax + b to solve the inequality.
Write the solution set in interval notation.
1
21) 4x - 11 ≥ x + 3
2
9
Provide an appropriate response.
31) The graphs of two linear functions f and g are
shown in the figure. Solve (i) the equation
f(x) = g(x) and (ii) the inequality g(x) < f(x).
y
8
7
6
5
(1, 4)
(4, 5)
4
3
2
1
-4 -3 -2 -1
-1
1
2
3
4
5
6
7
8x
Solve the inequality symbolically. Express the solution set
in interval notation.
22) a) 7a - 1 > 6a - 13
b) -3(5a - 3) ≤ -18a + 27
2x - 2
c)
<5
4
23) a) -3 < 10x - 7 ≤ 8
b) -7 ≤ 8 - 9x ≤ 16
Solve the inequality graphically. Express the solution in
set-builder notation.
24) 6x - 5 ≤ 5x + 4
Solve the equation.
25) r - 5 = 7
26) b + 6 - 8 = 0
27) 2m + 3 + 5 = 14
28) 4t - 3 + 9 = 5
Solve the absolute value equation.
29) |3x - 4| = |x + 5|
30) |x - 6| = |2 - x|
Solve the absolute value inequality. Write the solution set
using interval notation.
32) a) 9x + 5 < 16
b) b + 4 - 3 > 5
c) 3 x + 5 ≤ 6
33) b + 4 - 3 > 5
Solve the inequality graphically, numerically, or
symbolically, and express the solution in interval notation.
Where appropriate, round to the nearest tenth.
34) 3∣x + 5∣ < 6
Solve the problem.
35) The inequality |T - 48| ≤ 15 describes the
range of monthly average temperatures T in
degrees Fahrenheit at a City X.
i) Interpret the inequality.
ii) Solve the inequality.
Identify f as being linear, quadratic, or neither. If f is
quadratic, identify the leading coefficient a.
3x - 7
36) a) f(x) =
4
b) g(x) = 8 + 3x2
7
c) h(x) =
5x2 + 2
d) j(x) = -5x - 3
Math 1314, Review for Test 2
Page 3
Evaluate.
37) a) Given f(x) = 2x2 + 6x + 3, find f(-5).
b) Given f(x) = x2 - 4x + 3, find f(4).
The graph of f(x) = ax2 + bx + c is given in the figure. State
whether a > 0 or a < 0.
43) a)
10
Use the graph of the quadratic function to determine the
sign of the leading coefficient a, the vertex, and the
equation of the axis of symmetry. Then state the intervals
where f is increasing and where f is decreasing.
38)
5
4
3
2
1
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
5
-10
-5
10
x
5
10
x
-5
-10
b)
1 2 3 4 5 6x
10
y
5
-10
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
5
y
39)
5
4
3
2
1
y
-5
y
-5
-10
1 2 3 4 5 6 x
Use the given graph of the quadratic function f to write its
formula as f(x) = a(x - h)2 + k.
44)
4
3
2
1
Identify the interval where f is increasing or decreasing, as
indicated. Express your answer in interval notation.
40) a) f(x) = (x + 1)2 + 5
-3 -2 -1-1
-2
-3
-4
b) f(x) = -4x2 + 16x
c) f(x) = -6x2 + 12x - 10
Determine the vertex of the graph of f.
41) a) f(x) = (x - 1)2
b) f(x) = -2(x + 4)2 - 6
1
c) f(x) = (x + 4)2 + 3
5
d) f(x) = 2x2 - 20x + 53
e) f(x) = -3x2 + 24x - 43
Write the equation as f(x) = a(x - h)2 + k. Identify the
vertex.
42) f(x) = x2 + 6x - 4
y
1 2 3 4 5 6 7
x
45)
2
1
-4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
y
1 2 3 4 5 6 7 8 x
Math 1314, Review for Test 2
Find f(x) = a(x - h)2 + k so that f models the data exactly.
46)
x -3 -2 -1 0 1
y -3 3 5 3 -3
Page 4
The graph of f(x) = ax2 + bx + c is given in the figure.
Determine whether the discriminant is positive, negative,
or zero.
59)
y
10
Solve the quadratic equation. Give exact, simplified
answers, not decimals.
47) x2 - 8x + 15 = 0
5
48) 8x2 = 24x
-10
-5
5
10
x
5
10
x
-5
49) 5x2 = 65
-10
50) (p + 6)2 = 2
51) x2 + 10x + 25 = 13
60)
y
10
Solve by completing the square.
52) a 2 - 12a + 20 = 0
5
53) x2 + 4x = 7
-10
-5
-5
54) x2 = 5 - 6x
Use the discriminant to determine the number of real
solutions.
55) s2 - 4s - 5 = 0
56) t2 - 8t + 16 = 0
-10
The graph of f(x) = ax2 + bx + c is given in the figure. Solve
the equation ax2 + bx + c = 0.
61)
40
57) w2 - 4w + 6 = 0
y
30
20
58) (-1 + 4x)2 = 61
10
-10
-5
5
-10
-20
-30
-40
10
x
Math 1314, Review for Test 2
Page 5
62)
68) A grasshopper is perched on a reed 6 inches
above the ground. It hops off the reed and
lands on the ground about 10.4 inches away.
During its hop, its height is given by the
equation h = -0.2x2 + 1.50x + 6, where x is the
y
10
5
-10
-5
5
10 x
-5
-10
Graph the quadratic function. Identify the following:
i) the vertex,
ii) the axis of symmetry,
iii) the y-intercept,
iv) the x-intercept(s), if any.
63) a) f(x) = -x2 + 5
b) g(x) = -(x + 2)2 + 6
c) h(x) = x2 - 4x - 5
Solve the problem.
64) A ball is tossed upward. Its height after t
seconds is given in the table.
Time (seconds) 0.5
1
1.5
2
2.5
Height (feet)
26.5 39.5 44.5 41.5 30.5
Find a quadratic function to model the data.
Use the model to determine the following:
a) when the ball reaches its maximum height,
b) the maximum height. the ball reaches.
65) A farmer has 900 feet of fence with which to
fence a rectangular plot of land. The plot lies
along a river so that only three sides need to be
fenced. What is the largest area that can be
fenced.
66) A rock falls from a tower that is 400 feet high.
As it is falling, its height is given by the
formula h = 400 - 16t2 . How many seconds
will it take for the rock to hit the ground?
67) A ball is thrown downward from a window in
a tall building. Its position at time t in seconds
is s = 16t2 + 32t, where s is in feet. How long
(to the nearest tenth) will it take the ball to fall
226 feet?
distance in inches from the base of the reed,
and h is in inches. How far was the
grasshopper from the base of the reed when it
was 4.75 inches above the ground? Round to
the nearest tenth.
Simplify the expression using the imaginary unit i.
69) a) -64
b) -37
c) -212
Perform the indicated operation and write the expression
in standard form.
70) a) (2 - 7i) + (6 + 2i)
b) (9 + 8i) - (-6 + i)
c) 2i(4 - 5i)
d) (8 + 3i)(2 - 5i)
Multiply and write the result in standard form.
71) (7 - -4)(5 + -9)
Divide and write the result in standard form.
9 + 4i
72) a)
3 - 2i
b)
4 + 3i
5 + 2i
Solve the quadratic equation. Write complex solutions in
standard form.
73) a) x2 + x + 3 = 0
b) x2 - 8x + 52 = 0
c) x(5x + 3) = -2
Complete the following for the given f(x).
(i) Find f(x + h).
(ii) Find the difference quotient of f and simplify.
74) a) f(x) = 3x - 8
b) f(x) = 4x2 + 10x - 9
Find the difference quotient for the function and simplify
it.
75) a) f(x) = x2 - 8x
b) g(x) = 3x2 + 12x - 10
Math 1314, Review for Test 2
Page 6
Use the graph of f to determine the intervals where f is
increasing and where f is decreasing.
76)
4
y
3
2
1
-4
-3
-2
-1
1
2
3
4 x
-1
Provide the requested response.
82) For the function g(x) = 2x
-x + 6
a) Find g(-2)
b) Find g(0)
c) Graph the function
d) What is the domain of g?
Graph f. Use the graph to determine whether f is
continuous.
83)
if -2 ≤ x ≤ 0
-2x - 2
-2
-3
-4
Identify where f is increasing and where f is decreasing.
77) a) f(x) = 4x - 3
b) f(x) = x2 + 5
Solve the problem.
78) The distance D in feet that an object has fallen
after t seconds is given by D(t) = 16t2 .
(i) Evaluate D(2) and D(5).
(ii) Calculate the average rate of change of D
from 2 to 5. Interpret the result and include
appropriate units of measuer.
Write a formula for a linear function f that models the data
exactly.
2
4
79) x -4 -2 0
f(x) 22 14 6
-2 -10
Write a formula for a linear function f whose graph
satisfies the conditions.
80) Slope: 1.4; passing through (1, 3.3)
Solve the problem.
81) In a certain city, the cost of a taxi ride is
computed as follows: There is a fixed charge of
$2.85 as soon as you get in the taxi, to which a
charge of $1.85 per mile is added. Find an
equation that can be used to determine the
cost, C(x), of an x-mile taxi ride.
if -5 ≤ x ≤ 1
if 1 < x ≤ 5
f(x) =
-1
if 0 < x < 3
x -1
if 3 ≤ x ≤ 5
Answer Key
Testname: CAREVIEW2_F12
1) a) y = -3(x - 7) + 4
10
b) y = 8(x )+2
3
2 3
23) a) ( , ]
5 2
b) [-
2) y = -2x - 4
2
48
3) y = x +
5
5
4) y =
4
x+8
3
5) y =
4
22
x+
7
7
6) y =
7
x+5
8
24) {x x ≤ 9}
25) -2, 12
26) 2, -14
27) 3, - 6
28) No solution
9
1
29)
,2
4
30) 4
31) i) x = 1;
ii) {x|x < 1} or (-∞, 1)
7 11
32) a) - ,
3 9
7) x = -7
8) y = 5
9) 630 sec
y
10
5
-10
-5
5
-5
-10
10)
11) a) Linear
b) Nonlinear
c) Linear
5
12)
8
13)
1
7
14)
5
7
15)
29
66
8 5
, ]
9 3
16) All real numbers
17) a) Contradiction
b) Identity
c) Conditional
18) f(x) = 833x
19) 8 m
20) Width: 30 in.; length: 56 in.
21) [4, ∞)
22) a) (-12, ∞)
b) (-∞, 9]
c) (-∞, 11)
10
b) (-∞, -12) ∪ (4, ∞)
c) [-7, -3]
33) (-∞, -12) ∪ (4, ∞)
34) - 7, - 3
35) i) The monthly averages are
always within 15° of 48°F.
ii) {T| 33 ≤ T ≤ 63}
36) a) Linear
b) Quadratic; 3
c) Neither
d) Linear
37) a) 23
b) 3
38) a > 0
Vertex: (3, -2);
Axis of symmetry: x = 3
Increasing on [3, ∞)
Decreasing on (-∞, 3]
39) a < 0
Vertex: (-2, 3)
Axis of symmetry: x = -2
Increasing on (-∞, -2]
Decreasing on [-2, ∞)
40) [-1, ∞)
41) a) (1, 0)
b) (-4, -6)
c) (-4, 3)
d) (5, 3)
e) (4, -5)
42) f(x) = (x + 3)2 - 13; (-3, -13)
43) a) a > 0
b) a < 0
44) f(x) = (x - 3)2 - 3
45) f(x) = -2(x - 4)2 - 1
46) f(x) = -2(x + 1)2 + 5
47) 3, 5
48) 0, 3
49) ± 13
50) -6 ± 2
51) -5 ± 13
52) 10, 2
53) -2 ± 11
54) -3 ± 14
55) Two real solutions
56) One real solution
57) No real solutions
58) Two real solutions
59) Negative
60) Positive
61) 0, -9
62) 2
Answer Key
Testname: CAREVIEW2_F12
63) a)
i) vertex: (0, 5)
ii) axis of symmetry: x = 0
iii) y-intercept: (0, 5)
iv) x-intercepts: (-1, 0), (5, 0)
10
y
5
-10
-5
5
10 x
-5
-10
b)
i) vertex: (-2, 6)
ii) axis of symmetry: x = -2
iii) y-intercept: (0, 2)
iv) x-intercepts: (-2+ 6, 0),
(-2- 6, 0)
10
5
10 x
-10
c)
i) vertex: (2, -9)
ii) axis of symmetry: x = 2
iii) y-intercept: (0, -5)
iv) x-intercepts: (-1, 0), (5, 0)
y
5
-10
-5
5
6
4
2
-10 -8 -6 -4 -2
-2
-4
10 + 8i
31 - 34i
71) 41 + 11i
19 30
72)
i
+
13 13
1
±
2
10 x
-5
-10
64) The ball reaches a maximum
height of 44.6 feet in 1.6 seconds.
2
4
6 8 10
-6
-8
-10
d) domain: [-5, 5]
11
i
2
5
+ 10h - 9
(ii) 8x + 4h + 10
75) a) 2x + h - 8
b) 6x + 12 + 3h
76) increasing: [-2, 0] ∪ [1, ∞);
decreasing: (-∞, -2] ∪ [0, 1]
77) a) increasing: (-∞, ∞);
decreasing: never
b) increasing: [0, ∞); decreasing
(-∞, 0]
78) (i) D(2) = 64, D(5) = 400
(ii) 112; the object's average
speed from 2 to 5 seconds is 112
ft/sec.
79) f(x) = -4x + 6
80) f(x) = 1.4x + 1.9
81) C(x) = 2.85 + 1.85x
y
4
3
2
74) a) (i) 3x + 3h - 8
(ii) 3
b) (i) 4x2 + 8xh + 4h 2 + 10x
-5
10
8
b) 4 ± 6i
3
31
c) ±
i
10
10
y
-5
y
10
69) a) 8i
b) i 37
c) 2i 53
70) 8 - 5i
15 + 7i
73) a) -
5
-10
82) a) g(3) = 3
b) g(1) = 2
c)
65) 101,250 ft2
66) 5 sec
67) 2.9 sec
68) 8.3 inches
1
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
83)
Not continuous
2
3