#1: simplifying algebraic expressions · evaluate the algebraic expression for the given value ......
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John Jay College of Criminal Justice Name___________________________________The City University of New York, CUNY Fall 2017Department of Mathematics and Computer ScienceMAT 105 - College Algebra
Departmental Final Examination Review
Reminder: The Departmental Final Examination will take place on Wednesday, December 13. (Room and time will be setby the Registrar.)
Instructions: Answer all 121 questions on this review paper. As always, justify all your answers. Show your worksolving these questions with its corresponding number on a separate piece of paper. Answers without the accompanyingwork will not receive full credit.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Topic #1: Evaluating and Simplifying Algebraic Expressions
Evaluate the algebraic expression for the given value or values of the variable(s).
1) y - 7x6x + xy
; x = -2 and y = 3 1)
2) -b + b2 - 4ac2a
when a = 5, b = 14, and c = -3 2)
Simplify the algebraic expressions:3) (12y + 9) + (11y2 - 6y + 9) 3)
4) (3a - 2b - 5c) - (9a - 6b - 7c) 4)
5) (x - 11)(x2 + 7x - 5) 5)
6) -35x2 + 28x + 217 6)
Topic #2: Integer Exponents
Simplify the exponential expressions:7) (-6x4)(8x7) 7)
8) 20x9y11z9
4x4y3z88)
9) 25x13y6
5x3y3
09)
1
10) (-5x5y-6)(2x-1y) 10)
11) 21x13y13
7x12y-10 11)
Topic #3: Rational Exponents and Radicals
Evaluate the expression :12) 144 + 25 12)
Add or subtract terms whenever possible.13) 5 2 + 5 50 13)
14) 2x + 6 8x - 2 32x 14)
Rationalize the denominator.
15) 37 - 2
15)
Simplify the radical expression.
16)3x8 16)
Evaluate the expressions :17) 161/4 17)
18) 49-3/2 18)
Simplify by reducing the index of the radical.
19)20x16 19)
20)816x4 20)
Topic #4: Factoring
Factor out the greatest common factor.21) 21x4 - 6x3 + 15x2 21)
Factor by grouping.22) x3 + 9x - 3x2 - 27 22)
Factor the trinomial, or state that the trinomial is prime.23) x2 - 12x + 27 23)
2
24) 6x2 + 13x + 6 24)
Factor the difference of two squares.25) 49x2 - 16y2 25)
Factor using the formula for the sum or difference of two cubes.26) 64x3 - 1 26)
27) 125x3 + 1 27)
Topic #5: Rational Expressions
Perform the indicated operations and simplify the result. Leave the answer in factored form.
28) 4x - 4x
∙ 8x2
5x - 5 28)
29) x2 - 10x + 24x2 - 21x + 108
∙ x2 - 14x + 24x2 - 16x + 60
29)
Add or subtract as indicated.
30) 4x2 - 3x + 2
+ 5x2 - 1
30)
31) xx2 - 16
- 5x2 + 5x + 4
31)
Topic #6: Complex Numbers
Add or subtract as indicated and write the result in standard form.32) -7 - (- 2 - 7i) - (- 2 + 5i) 32)
Find the product and write the result in standard form.33) (-3 - 7i)(2 + i) 33)
Divide and express the result in standard form.
34) 84 + i 34)
35) 6 - 6i8 + 2i 35)
Perform the indicated operations and write the result in standard form.36) -16 + -81 36)
3
37) -2 - -242 37)
Topic #7: Linear, Rational, Radical, Absolute Value, and Literal Equations
Solve and check the linear equations.38) (-5x + 4) - 5 = -4(x - 7) 38)
39) 2x5 = x3
+ 5 39)
First, write the value(s) that make the denominator(s) zero. Then solve the equation.
40) 10x = 52x
+ 30 40)
Solve the absolute value equation or indicate that the equation has no solution.41) 3 x - 3 = 18 41)
Solve the radical equation, and check all proposed solutions.42) 6x + 55 = x 42)
Solve the formula for the specified variable.
43) F = 95C + 32 for C 43)
44) A = 12bh for b 44)
45) S = 2!rh + 2!r2 for h 45)
46) P = 2L + 2W for W 46)
Topic #8: Linear, Compound, and Absolute Value Inequalities
Solve the linear inequality. Other than ∅, use interval notation to express the solution set and graph the solution set ona number line.
47) 7x - 6 ≥ 6x - 2 47)
48) -8x + 4 ≤ -2(3x + 1) 48)
4
Solve the compound inequality. Other than ∅, use interval notation to express the solution set and graph the solutionset on a number line.
49) 17 ≤ 5x - 3 ≤ 22 49)
Solve the absolute value inequality. Other than ∅, use interval notation to express the solution set and graph thesolution set on a number line.
50) |x + 2| + 6 ≤ 11 50)
51) |7x - 9| - 3 > -6
2 4 6 8 10 12 142 4 6 8 10 12 14
51)
52) |5x - 8| - 9 < -14
1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9
52)
Topic #9: Distance and Midpoint Formulas; Circles
Find the distance between the pair of points.53) (-1, 4) and (-5, 7) 53)
Find the midpoint of the line segment whose end points are given.54) (7, 3) and (4, 1) 54)
Write the standard form of the equation of the circle with the given center and radius.55) (-4, 4); 3 55)
Find the center and the radius of the circle.56) (x - 5)2 + (y + 7)2 = 36 56)
Complete the square and write the equation in standard form. Then give the center and radius of the circle.57) x2 - 12x + 36 + y2 - 8y + 16 = 16 57)
5
Graph the equation.58) (x - 1)2 + (y - 2)2 = 49
x-10 -5 5 10
y
5
-5
x-10 -5 5 10
y
5
-5
58)
Topic #10: Basics of Functions and Their Graphs
Determine whether the relation is a function.59) {(-7, -1), (-7, 2), (-1, 8), (3, 3), (10, -7)} 59)
Evaluate the function at the given value of the independent variable and simplify.60) f(x) = -3x - 8; f(-2) 60)
61) f(x) = x + 11; f(-2) 61)
Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.62)
x
y
x
y62)
6
63)
x
y
x
y63)
64)
x
y
x
y64)
Use the graph to find the indicated function value.65) y = f(x). Find f(-1)
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
65)
7
Use the graph to determine the function's domain and range.66)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
66)
67)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
67)
Identify the intervals where the function is changing as requested.68) Increasing
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
68)
8
69) Constant
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
69)
Evaluate the piecewise function at the given value of the independent variable.70) f(x) = 3x + 1 if x < -1
-2x - 5 if x ≥ -1; f(2) 70)
Determine whether the given function is even, odd, or neither.71) f(x) = x3 - 5x 71)
72) f(x) = 2x2 + x4 72)
73) f(x) = x3 - x2 73)
Topic #11: Slope and Linear Functions
Find the slope of the line that goes through the given points.74) (-2, -6), (-9, -17) 74)
Use the given conditions to write an equation for the line in point-slope form.75) Slope = 4, passing through (-3, 7) 75)
Use the given conditions to write an equation for the line in slope-intercept form.
76) Slope = 23, passing through (7, 3) 76)
77) Passing through (-8, -2) and (-5, -7) 77)
9
Graph the line whose equation is given.78) y = 2x - 2
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
78)
Determine the slope and the y-intercept of the graph of the equation.79) 7x - 10y - 70 = 0 79)
Use the given conditions to write an equation for the line in the indicated form.80) Passing through (2, 3) and parallel to the line whose equation is y = -2x + 3 ;
point-slope form80)
81) Passing through (5, 3) and perpendicular to the line whose equation is y = 2x + 7;point-slope form
81)
Find the average rate of change of the function from x1 to x2.
82) f(x) = -3x2 - x from x1 = 5 to x2 = 6 82)
Find and simplify the difference quotient of f, f(x + h) - f(x)h
, h≠ 0, for the function.
83) f(x) = 4x2 83)
Topic #12: Transformations of Graphs
Begin by graphing the standard quadratic function f(x) = x2 . Then use transformations of this graph to graph thegiven function.
84) h(x) = (x - 7)2 - 5
x-10 -8 -6 -4 -2 2 4 6 8 10
y
8
6
4
2
-2
-4
-6
-8
x-10 -8 -6 -4 -2 2 4 6 8 10
y
8
6
4
2
-2
-4
-6
-8
84)
10
Use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g.85) g(x) = -f(x - 1) + 2
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
y = f(x)
85)
Topic #13: Algebra of Functions, Function Composition, and Inverse Functions
Given functions f and g, perform the indicated operations.86) f(x) = 3 - 5x, g(x) = -8x + 5
Find f + g.86)
For the given functions f and g , find the indicated composition.87) f(x) = 3x + 9, g(x) = 5x - 1
(f∘g)(x)87)
88) f(x) = x2 + 2x + 2, g(x) = x2 - 2x - 3(f∘g)(-3)
88)
The function f is one-to-one. Find its inverse.89) f(x) = 3x - 7 89)
90) f(x) = x + 7 90)
91) f(x) = 3x - 78x + 4
91)
Topic #14: Quadratic Equations and Quadratic Functions
Solve the equation by factoring.92) x2 = x + 6 92)
Solve the equation by factoring.93) x2 + 2x - 120 = 0 93)
Solve the equation by the square root property.94) 6x2 = 54 94)
11
95) (x - 3)2 = 49 95)
Solve the equation by completing the square.96) x2 + 14x + 26 = 0 96)
Solve the equation using the quadratic formula.97) x2 + 7x + 7 = 0 97)
98) 5x2 - 3x + 3 = 0 98)
The graph of a quadratic function is given. Determine the function's equation.99)
x-10 -8 -6 -4 -2 2 4 6 8 10
y108642
-2-4-6-8-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y108642
-2-4-6-8-10
99)
100)
x-10 -8 -6 -4 -2 2 4 6 8 10
y108642
-2-4-6-8-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y108642
-2-4-6-8-10
100)
Find the coordinates of the vertex for the parabola defined by the given quadratic function.101) f(x) = (x - 4)2 - 4 101)
102) y + 4 = (x - 2)2 102)
Find the axis of symmetry of the parabola defined by the given quadratic function.103) f(x) = x2 + 7 103)
104) f(x) = (x + 4)2 - 6 104)
12
Topic #15: Introduction to Polynomial and Rational Functions
Form a polynomial whose zeros and degree are given.105) Zeros: -3, -2, 2; degree 3 105)
For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axisat each x -intercept.
106) f(x) = 5(x + 3)(x - 3)3 106)
107) f(x) = 2(x2 + 4)(x + 1)2 107)
Find the x- and y-intercepts of f.108) f(x) = (x + 4)(x - 2)(x + 2) 108)
109) f(x) = 4x - x3 109)
List the potential rational zeros of the polynomial function. Do not find the zeros.110) f(x) = 6x4 + 2x3 - 3x2 + 2 110)
Use the Remainder Theorem to find the remainder when f(x) is divided by x - c.111) f(x) = x4 + 8x3 + 12x2; x + 1 111)
Form a polynomial f(x) with real coefficients having the given degree and zeros.112) Degree 3: zeros: 1 + i and -5 112)
Use the given zero to find the remaining zeros of the function.113) f(x) = x4 - 21x2 - 100; zero: -2i 113)
Divide using synthetic division.
114) x4 - 3x3 + x2 + 4x - 5
x - 1 114)
Use the Leading Coefficient Test to determine the end behavior of the polynomial function.115) f(x) = 3x4 + 4x3 - 4x2 + 3x - 2 115)
116) f(x) = 2x3 + 5x2 + 5x + 5 116)
Find the domain of the rational function.
117) g(x) = 2xx + 2
117)
118) f(x) = x + 7x2 - 9
118)
13
119) f(x) = x + 2x2 + 16x
119)
Find the vertical asymptotes of the rational function.
120) h(x) = 4x2(x + 2)(x - 6)
120)
121) g(x) = x + 4x2 + 4
121)
14
Answer KeyTestname: MAT105_DFE_REVIEW_TEMPLATE_(SA)
1) - 1718
2) 15
3) 11y2 + 6y+ 184) -6a + 4b + 2c5) x3 - 4x2 - 82x + 556) -5x2 + 4x + 37) -48x11
8) 5x5y8z9) 1
10) -10x4
y5
11) 3xy2312) 1713) 30 214) 5 2x
15) 21 + 3 247
16) x23x2
17) 2
18) 1343
19)5x4
20) 2x21) 3x2(7x2 - 2x + 5)22) (x - 3)(x2 + 9)23) (x - 9)(x - 3)24) (3x + 2)(2x + 3)25) (7x + 4y)(7x - 4y)26) (4x - 1)(16x2 + 4x + 1)27) (5x + 1)(25x2 - 5x + 1)
28) 32x5
29) (x - 4)(x - 2)(x - 9)(x - 10)
30) 9x - 6(x - 1)(x + 1)(x - 2)
31) x2 - 4x + 20(x - 4)(x + 4)(x + 1)
15
Answer KeyTestname: MAT105_DFE_REVIEW_TEMPLATE_(SA)
32) -3 + 2i33) 1 - 17i
34) 3217 - 817 i
35) 917 - 1517 i
36) 13i37) -1 - i 638) {- 29}39) {75}
40) x ≠ 0; 14
41) {9, -3}42) {11}
43) C = 59(F - 32)
44) b = 2Ah
45) h = S - 2!r22!r
46) W = P - 2L2
47) [4, ∞)
-2 -1 0 1 2 3 4 5 6 7 8 9 10-2 -1 0 1 2 3 4 5 6 7 8 9 10
48) [3, ∞)
-3 -2 -1 0 1 2 3 4 5 6 7 8 9-3 -2 -1 0 1 2 3 4 5 6 7 8 9
49) [4, 5]
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11-2 -1 0 1 2 3 4 5 6 7 8 9 10 11
50) [-7, 3]
-12 -10 -8 -6 -4 -2 0 2 4 6 8-12 -10 -8 -6 -4 -2 0 2 4 6 8
51) (-∞, ∞)
2 4 6 8 10 12 142 4 6 8 10 12 14
52) ∅
1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9
53) 5
16
Answer KeyTestname: MAT105_DFE_REVIEW_TEMPLATE_(SA)
54) ( 112 , 2)
55) (x + 4)2 + (y - 4)2 = 956) (5, -7), r = 657) (x - 6)2 + (y - 4)2 = 16
(6, 4), r = 458)
x-10 -5 5 10
y
5
-5
x-10 -5 5 10
y
5
-5
Domain = (-6, 8), Range = (-5, 9)59) Not a function60) -261) 362) not a function63) not a function64) function65) 4.266) domain: (-∞, ∞)
range: [-4, ∞)67) domain: [0, ∞)
range: [-1, ∞)68) (-2, 2)69) (-1, 1)70) -971) Odd72) Even73) Neither
74) 11775) y - 7 = 4(x + 3)
76) y = 23x - 5
3
77) y = - 53 x - 46
3
17
Answer KeyTestname: MAT105_DFE_REVIEW_TEMPLATE_(SA)
78)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
79) m = 710 ; (0, -7)
80) y - 3 = -2(x - 2)
81) y - 3 = - 12 (x - 5)
82) -3483) 4(2x+h)84)
x-8 -6 -4 -2 2 4 6 8
y
8642
-2-4-6-8
x-8 -6 -4 -2 2 4 6 8
y
8642
-2-4-6-8
85)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
86) -13x + 887) 15x + 688) 170
89) f-1(x) = x + 73
18
Answer KeyTestname: MAT105_DFE_REVIEW_TEMPLATE_(SA)
90) f-1(x) = x2 - 7, x ≥ 0
91) f-1(x) = -4x - 78x - 3
92) {-2, 3}93) {-12, 10}94) {-3, 3}95) {-4, 10}96) {-7 - 23 , -7 + 23}
97) -7 - 212
, -7 + 212
98) 3 ± i 5110
99) f(x) = (x + 2)2 + 2100) j(x) = -x2 + 1101) (4, -4)102) (2, - 4)103) x = 0104) x = -4105) f(x) = x3 + 3x2 - 4x - 12 for a = 1106) -3, multiplicity 1, crosses x-axis; 3, multiplicity 3, crosses x-axis107) -1, multiplicity 2, touches x-axis108) x-intercepts: -4, -2, 2; y-intercept: -16109) x-intercepts: 0, 2, -2; y-intercept: 0
110) ± 16, ± 13, ± 12, ± 23, ± 1, ± 2
111) R = 5112) f(x) = x3 + 3x2 - 8x + 10113) 2i, 5, -5
114) x3 - 2x2 - x + 3 - 2x - 1
115) rises to the left and rises to the right116) falls to the left and rises to the right117) {x|x ≠ -2}118) {x|x ≠ -3, x ≠ 3}119) {x|x ≠ 0, x ≠ -16}120) x = -2, x = 6121) none
19