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Pre-Calculus Summer Review Packet
Name ______________________________
1.
Solve the inequality: 4x  5  2(x  3)
2.
Solve the inequality and state the answer in interval notation: a) 4x < 16 or 12x > 144
3.
Solve the equations:
a. x  3
e.  4  8b  12
4.
b.
f.
x  3
c.
3 x4 9
d.
b) 11  3y  2  20
2x  5  3
2 4t  1  6  20
Eric pays $363 in advance on his account at Bally Total Fitness. Each time he uses the club, $6.00 is deducted
from the account. Write an equation that represents the value remaining in his account after x visits to the club.
Find the value remaining in the account after 14 visits.
6.
6
x 5.
5
Find the slope and the y-intercept of the line 4x  6y  6 .
7.
Write the equation of a line in point-slope form, slope-intercept form and standard form that passes through the
5.
Graph the equation y 
points: (3, -6) and (10, -9).
8.
Write the equation of a line in slope-intercept form that passes through the point (3, -5) and has a slope m=8.
9.
Write the equation of a line in slope-intercept form that passes through the point (0,-4) and is parallel to the line
y  5x  2 .
10.
Write the equation of a line in slope-intercept form that passes through the point (8, -9) and is perpendicular to
the line 7x  2y  2 .
11.
Graph y  x  2  5
12.
Graph y< -3x + 4
13.
Write an inequality with y related to x for the graph:
14.
Graph y   x  7
15.
Discuss the transformation to the graph y   x  7  6 from y  x
16.
17.
18.
19.
x  y  4
Use a graph to solve the system of equations and classify the system 
 y  2x  5
2x  y  14
3y  6
Solve by substitution & classify:
a. 
b. 
6x  4y  38
 y  2x  6
9x  2y  19
a. 
 4x  y  9
 x  0 and y  0

Graph the system of linear inequalities: 5x  3y  15
6y  2x  18

Solve by elimination & classify:
b.
2x  3y  7

8x  9y  7
20.
Aaron, an analyst at Mega Corp, has developed a model to forecast the company’s profits from investments.
Under the model, profit P, is equal to 0.26 times the investment in equipment plus 0.37 times the investment in
labor. Budget constraints mean that total investment in labor and equipment cannot exceed $245,000; logically,
investment in labor and equipment cannot be negative. Write the objective function for profit, P, and the set of
constraints to which it applies. Graph the set of constraints. Let x = Labor and y = Equipment.
21.
Graph the feasible region for the set of constraints, and find the maximum and minimum values, if they exist, of
the objective function C= 2x +5y.
4x  2y  12
5y  x  15
Constraints: x  y  3
22.
23.
24.
25.
26.
Solve:
x  y  z  2

 x  2z  5
2 x  y  z   1

Is the function quadratic? If so, identify the quadratic, linear, and constant term: f(x) = -(x-4) (x+2) -3x
Graph the function. Identify the vertex and axis of symmetry. f(x) = 4x 2 – 12x + 9
Graph the function. Identify the vertex and axis of symmetry. f(x) = -4(x+8)2 +8. List the transformations that
occurred to the function.
Factor.
c. x 2  5x  24
a. x 2  9x  36
b. 5x 2  13x  6
d) 16x2 – 49
e) -15x3 - 20x
27.
Simplify.
28.
29.
Find the additive inverse and absolute value of: 3i - 7
Solve for x.
a. 4x 2  20
b. (x  5)2  16
c. x 2  5x  6  18
a.
(4  2i)  (7  9i)
e. 4x  4x 2  7
30.
31.
Complete the square.
a. x 2  20x 
33.
34.
(6  i)  (4  3i)
(5  2i)(8  5i)
c.
d.
x 2  10x  5
3x 2  2x  6  0
f.
b.
x 2  5x 
Rewrite in vertex form. Does the parabola open up or down?
a. f(x)  x 2  12x  3
b. f(x)  x 2  2x  10
c.
32.
b.
f(x)  3x 2  18x  20
d.
f(x)  5x 2  20x  13
Find the discriminant. State the number of real solutions.
a. x 2  10x  25  0
b.  2x 2  3x  16  0
c.
Find the zeros (or x-intercepts).
a. f(x)  9x 2  24x  16
b.
2x 2  5x  3  0
d.
f(x)  6x 2  2x  1
Identify the type of conic.
a. x2 + y2 – 8x + 6y – 56 = 0
b.
9x2 – y2 – 90x + 4y + 302 = 0
c.
d.
25x2 + 4y2 – 150x + 32y + 189 = 0
x = -5y2 + 30y + 11
4x 2  4x  9  0
35.
Identify the vertex, focus and directrix for the parabola: y = 1/12(x + 3)2 + 4
36.
Identify the center and radius for the circle: (x – 7)2 + (y + 1)2 = 16
37.
The graph of (x  1)2  (y  2)2  4 is translated left 4 units and down 3 units. Write the new equation.
38.
Solve for x and y.
2x 2  5y 2  98
a.
2x 2  y 2  2
b.
x 2  5y 2  81
2x 2  4y 2  66
c.
x2  y2  1
x2  y2  4
39.
Use the data to complete the following.
a. Display the data in a matrix with the types of unemployment in the columns.
b. State the dimensions of the matrix.
c. Identify a21, and tell what it represents.
d. Identify a16, and tell what it represents.
40.
Find each sum or difference.
a.
41.
b.
Find the value of each variable.
a.
42.
b.
Solve each matrix equation.
a.
43.
Use the following matrices to find each product, sum, or difference, if possible. If not possible, write undefined.
a. 3AB
44.
b.
b. 2A + 4D
b.
Solve each matrix equation for X.
a.
46.
d. DA
Evaluate for the determinant of each matrix.
a.
45.
c. 2C – E
b.
Solve each system of equations.
a.
b.
47.
a.
48.
Classify each of the polynomials by degree and number of terms. Describe the shape of its graph.
b.
x(x  5)  5(x  5)
(x  2)(x  4)(x  6)
Find a cubic function to model the data below. (Hint: Use the number of years past 1940 for x.) Then use the
function to estimate the average monthly Social Security Benefit for a retired worker in 2010.
Average Monthly Social Security Benefits, 1940-2003
Year
1940
1950
1960
1970
1980
1990
2000
2003
Amount ($)
22.71
29.03
81.73
123.82
321.10
550.50
844.60
922.10
49.
Determine the end behavior of the graph of the function f(x)  5x 6  x 4  3x 3 .
50.
Graph the function and approximate any local maxima or minima to the nearest tenth.
P(x)  2x5  2x 4  3x 2  8x  3
51.
Write a polynomial in standard form with zeros at x = -1 (multiplicity 2) and
52.
Factor each polynomial completely:
a. x 3  1
b. 125x 3  27
c.
x3  5x2  x  5
d.
5
x4  8x2  15
53.
Divide the polynomials: (3x 4  3x3  6x  12)  (x  2)
54.
Use synthetic division and the remainder theorem to find P(3) for the polynomial P(x)  2x3  4x 2  10x  9 .
55.
Find a third-degree polynomial equation with rational coefficients that has roots of -2i and 6.
56.
For each equation, find the number of complex roots, the possible number of real roots, and the possible rational
roots, and the actual roots:
a.
2x4  x3  22x2  48x  32  0
57.
Simplify: a)
58.
3 3 5y 3 2 3 50y 4
59.
60.
61.
4
x 8 y 12
b.
b)
3
x3  x 2  x  2  0
81x 3y 6
5 2
3 7x
(3  4 2)(5  6 2)
5 3
2 3
1
62.
(32y 15 ) 5

3
4
63.
 625 
 256 


64.
(64a9 ) 3
65.
Solve:
66.
Solve:
67.
Find the inverse:
4
3
x 1 1  4
x  4x  5  4
68.
y  x  2 Is the inverse a function?
Graph. List transformations. Find domain and range. y  3 x  2
69.
Given f(x) = x2 + 1 and g(x) = 3x -2 find a) f(g(3))
70.
71.
72.
Write an exponential function y = a bx for a graph including points (4, 8) and (6, 32)
Graph y = 3 (.5)x
A population of a United States was 248,718,301 in 1990 and was projected to grow at a rate of about .8% per
year. Find the growth factor, and write a function to model the growth.
b)
f g  x
73.
Tell whether each function represents exponential growth or decay: y  5(1.2) x and y  10(0.8) x
74.
Describe the transformations of each graph from the graph of y  2 x
75.
Write in logarithm form
a)
82 = 64
76.
Evaluate:
a)
log 4
77.
Write in exponential form: a)
78.
Write log2 9 – log2 3 as a single logarithm
79.
Expand the logarithm: log 3m 4n-2
80.
Identify all excluded values, asymptotes, and holes in the graph of this rational function.
x2  9
g(x) = 2
x  10x  21
x 1
x2  x  6
Simplify:
.
 2
2
x  2x  3 x  2x  3
x 2  5x  6 x  2
Simplify:

x 2
x2  4
81.
82.
83.
Simplify:
84.
Solve:
y   2
x 1
x
b.
1
y   5
2
10-2 = .01
b)
1
x
256
log2 128 = 7
a.
b)
b)
log49 7 = x
log4 1 = 0
12
3

x 3 x 2
b
b
10

 2
b3 b2 b b6
x2
3
x2
9x 2 y 3xy 5
86. Simplify:

2xy 3
8y
85.
Solve:
x
1
4
x 2  16
87. Simplify: 2x  8
88.
The first five terms of a sequence are 15, 8, 1, -6, and –13. Write the next 3 numbers in the sequence.
89.
Find the 10th term of the arithmetic sequence in which t3 = 15 and t6 = 39.
90.
Find five arithmetic means between 7 and 43.
91.
Find two geometric means between 64 and 125.
92.
Write an explicit formula for the sequence 16, 24, 36, 54,…
93.
Find S20 of the arithmetic series 9, 2, -5, -12, -19 …
94.
Given 4, 12, 20, 28,… find S40.
95.
Evaluate. Round to nearest tenth if needed. a)
9
 .5(4k
k
1
1
)
b)
8
 (3n  2)
n 1
Answers:
1.
1
2
(, 4) U [12, )
x
2. a)
b)
(3, 6]
3. a. x=3, -3; b. no solution; c. x = 7 or x = 1; d. x = -1 or x = -4
4. Value  363  6x;$279
5.
6. m 
2
,b  1
3
3
3
(x  3) or y  9  (x  10)
7
7
3
33
SI y   x 
7
7
Std 3x  7y  33
PS y  6 
7.
8.
9.
y  8x  29
y  5x  4
2
47
10. y   x 
7
7
11.
12.
Shade here
13. y  2x  2
14.
15. shift 7 left, V Ref (over x), V shift 6 down
e.
(, )
f.
3
(,  ) U (2, )
2
16. (3, 1)
17. a. (9, 4); b. (4, 2)
18. a. (1, -5); b. (7, -7)
19.
20. Objective function: P = 0.26E + 0.37L
L  0; E  0
E + L  $245,000
21. The maximum value of C is at (5,4) and the minimum value is at (3,0)
22. x = 1, y = -1, z =2
23. Yes, Q -x2 L -x C 8
AS x = -8, H shift 8 left, V stretch 4, V Ref
(y-axis), V shift up 8
24. Vertex (3/2, 0) AS x = 3/2
25. Vertex (-8, 8)
26.
b.
c.
d.
e.
a. (x – 12)(x + 3)
(5x + 3)(x + 2)
(x – 8)(x + 3)
(4x -7) (4x + 7)
-5x (3x2 + 4)
27. a. –3 + 7i
b. 2 + 4i
c. 30 – 41i
28. additive inverse: -3i + 7 absolute value: 58
29.
a. x   5
b. x  9,1
c. x  8, 3
25
30. a. 100
b. 6.25 or
4
31.
d. x  5  2 5
e.
1
a. f(x)  (x  6)2  39; opens up
b. f(x)  (x  1) 2  11; opens up
c. f(x)  3(x  3)2  7; opens up
d. f(x)  5(x  2)2  7; opens down
32. a. 0; 1 Real zero w/a multiplicity of 2
4
3
34. a.
Circle: (x – 4)2 + (y + 3)2 = 81
c.
f.
c. 1; 2 Real zeros
1
17 i
3
d. -128; 0 Real
1 7
1 7 1 7
or x 
,
6
6
6
33. a. x 
b. x 
b. 137; 2 Real zeros
6 i
2
2
Parabola x = -5(y – 3) + 56
b. Hyperbola
d.
Ellipse
 y  2
2

81
 x  3
4
2
 x  5
1
9
y  4

25
35. vertex ( -3, 4) focus (-3,7) directrix (y = 1)
36. center (7, -1) radius = 4
37. (x + 3)2 + (y + 5)2 = 4
38. a. (3,4), (3,-4), (-3, 4), (-3, -4) b. (1,4), (-1,4), (1,-4), (-1,-4)
2
c. No Solution
2
1
17.6 8.3 5.4 8.7 4.0 6.6 3.5 
 9.5 5.1 4.5 6.4 2.6 5.1 2.7 

39. a) 
b) 2x7
c) 9.5 % unemployment in construction in June 1996
1992
d) 6.6 % unemployment in services in June
40.
 8  5 6 
 3 8 12 


 4 12 18
a)
41. a)
 7 
 4 
 
 23 
b)  
a = -3; b = 0; c = 5/3; x = 4; y = -7; z = 7/2
 1 4 
42. a) 

0 4
 8 11


b) 13 14


 4 11
43. a) not possible
b)
44. 6.
b) 39
a) 36
 6 2 
6 0 


c) not possible d)
b) x = 3; z = -2
1  1 
 3 2 


 3 
1
b)
 
 2 
3
45. a)  
4
 1
 75 
10.7 
7
 


46. a) 
 or  22  b)  7 

3.1


 3
7

47. a) Quadratic binomial b) cubic polynomial
48. y = 8.80x3 + .22x2 – 3.15x + 29.05
x   y  
49.
50.
51.
52.
x   y  
Max @ y= 4.148, Min @ y = -8
f(x) = x4 +2x3 – 4x2 -10x -5
a) (x + 1) (x2 – x + 1) b) (5x – 3) ( 25x2 + 15x + 9)
3
2
53. 3x  3x  6x  6
54. 51
55. f(x) = x3 – 6x2 + 4x – 24
c) (x + 5) (x – 1) ( x + 1)
5  i 7
1
, 2, 4, 8, 16, 32 ; 2, 4,
4
2
1i 3
b) 3 roots; 1 or 3 real roots; 1, 2 ; 2,
2
56. a) 4 roots; 0, 2, or 4 real roots; 1, 
57. a) x 2 y 3
58. 30y 2 3 2y
59.
5 14 x
21x
60. 63  38 2
61. 13  7 3
b)
3xy 2 3 3
d) (x2 +5) (x2 + 3)
62. -2y3
64
63.
125
64.
256a12
65. X = 28
66. x = 11, 1 ext
67. y  x 2  2 , yes
68. Domain: [0, ) Range: (, 2] Reflection, V stretch of 3, V shift 2 up
69. a)
50
b)
9x2 -12x + 5
70. y = .5 (2)x
71.
72.
73.
74.
75.
76.
77.
78.
79.
1.008; y = 248,718,301 (1.008)x
growth; decay
a. reflect across x-axis, shift left
a.2 = log8 64 b. -2 = log .01
7
a. 
b. ½
2
a. 128 = 27
b. 1 = 40
log2 3
log 3 + 4 log m – 2 log n
b. reflect across y-axis, shift down 5
80. excluded values: x = -3 and x = -7; hole x = -3; vertical asym.: x=-7; horizontal asym: y = 1.
81.
x 2
 x  1 x  3 
82.
x3
x2
83.
9x  33
x2  x  6
84. b = -2
85. (, 4)
86.
12
y6
87.
1
2
(2, )
88. -20, -27, -34
89. t10 = 71
90. 13, 19, 25, 31, 37
91. 80, 100
92. tn = 16 (1.5)n – 1
93. -1150
94. S20 = 6400
95. a) 43,690.5
b) -124