Final Exam Review - Lecture notes Lectures 1 - 25 PDF

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This is a final exam review of all topics covered in MAT 105....

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John Jay College of Criminal Justice The City University of New York, CUNY Department of Mathematics and Computer Science

FINAL DEPARTMENTAL EXAMINATION REVIEW MAT 105 College Algebra

1.Evaluating and Simplifying Algebraic Expressions: Evaluate the algebraic expression for the given value or values of the variable(s). y - 7x ; x = -2 and y = 3 1) 6x + xy 18

2)

-b +

B)

11 6

C) - 1

1) D)

11 18

b 2 - 4ac when a = 5, b = 14, and c = -3 2a 1 5

B) -

1 5

2) C) -3

D) 3

Simplify the algebraic expressions: 3) (12y + 9) + ( 11y 2 - 6y + 9) A) 11y 2 + 18y - 18

3) B) 29y 6

C) 11y 2 - 6y + 18 4) (3a - 2 b - 5c) - (9a - 6b - 7c) A) -6a + 4b + 2c B) 12a - 8 b - 12c

4) D) -6a - 8 b + 2c

5) (x - 11)(x2 + 7x - 5) C) x3 + 18x2 + 72x - 55

5) B) x3 + 18x2 + 82x + 55 D) x3 - 4x2 - 72x - 55

1

6)

-35x2 + 28 x + 21 7

6) B) -35x2 + 28x + 3 D) -5x2 + 28 x + 21

C) -245x2 + 196 x + 147

2. Exponential Expressions. Simplify the exponential expressions: 7) (-6x4)(8x7)

7)

A) -48x28

8)

C) 48x11

20x9y 11z9 4x4y 3z8

8)

A) 5x4y 7z 25x13y 6 5x3y 3

9)

B) 5x5y 8

D) x5y 8z

0 9)

A) x10y 3

C) 1

10) (-5x5y -6)(2x-1y) -10x6 A) y7

11)

D) 48x28

D) 0 10)

B)

-3x4 y5

D) -10x4y 7

y5

21x13y 13 7x12y -10

11) B) 3x25y 23

C) 3xy 23

D) 21xy 23

3. Radicals and Rational Exponents: Evaluate the expression : 12)

144 + 25 A) 13

12) B) 169

D)

119

Add or subtract terms whenever possible. 13) 5

2+5

50

13) B) 30

14)

2x + 6 A) 5

8x - 2 42x

2

C) -30

2

D) 20

2

32x

14) B) 4

42x

D) 4

2x

2

Rationalize the denominator. 3 15) 7- 2 A)

21 + 3 5

2

15) B)

3 3 7 2

C)

21 - 3 47

2

)

21 + 3 47

2

Simplify the radical expression. 3 16) x8 A) x2

3

x

16) B) x

3

x

x2

D) x

3

x2

Evaluate the expressions : 17) 161/4 A) 8

17) B) 16

C) 32

18) 49-3/2 A)

18) 1 343

)

C) 343

Simplify by reducing the index of the radical. 20 19) x16 4 5 A) x4 B) x

20)

8

D) -

1 343

19) 5

x4

D)

4

x

16x4 A) 2 2x

20) B)

1 4x

4

2x

D)

2x

4. Factoring Polynomials: Factor out the greatest common factor. 21) 21x4 - 6x3 + 15x2

21)

A) 3(7x4 - 2x3 + 5x2) C) 3x(7x3 - 2x2 + 5x) Factor by grouping. 22) x3 + 9 x - 3x2 - 27 A) (x - 3)(x2 + 9)

B) x2(21x2 - 6x + 15) 3x2(7x2 - 2x + 5)

22) B) (x - 3)(x + 9)

C) (x - 3)(x2 - 9)

D) (x + 3)(x2 + 9)

Factor the trinomial, or state that the trinomial is prime. 23) x2 - 12 x + 27 A) (x + 9)(x - 3)

B) (x + 9)(x + 1)

23) D) prime

3

24) 6x2 + 13 x + 6 A) (6x + 2)(x + 3) C) (3x + 2)(2x + 3) the

of 49x2 A) (7x + 4y)2

B) 7x + 4y)(7x - 4 y)

C

D) prime

-

Solve and check the linear equations. 26) (-5x + 4) - 5 = -4(x - 7) A) {19}

27)

26) C) {- 6}

D) {29}

2x x = +5 5 3 ) {-75}

27) B) {150}

C) {75}

D) {-150}

Solve the linear inequality. Other than ∅, use interval notation to express the solution set and graph the solution set on a number line. 28) 7x - 6 ≥ 6x - 2 28)

) [4, ∞)

B) (-8, ∞)

C) (-∞, 4]

D) (-∞, 4)

4

29) -8x + 4 ≤ -2(3x + 1)

29)

[3, ∞)

B) (-∞, 3)

C) (3, ∞)

D) (-∞, 3]

Solve the compound inequality. Other than ∅, use interval notation to express the solution set and graph the solution set on a number line. 30) 17 ≤ 5x - 3 ≤ 22 30)

A) (4, 5)

B) (-5, -4)

) [4, 5]

D) [-5, -4]

5

Solve the absolute value inequality. Other than ∅, use interval notation to express the solution set and graph the solution set on a number line. 31) 31) |x + 2| + 6 ≤ 11

A) [-7, 11]

B) (-7, 3)

[-7, 3]

D) (-∞, -7] ∪ [3, ∞)

32) |7x - 9| - 3 > -6

32)

) (-∞, ∞)

B)

6 12 , 7 7

C)

6 ,∞ 7

D) ∅

Solve the formula for the indicated letter. 9 33) F = C + 32 for C 5 C = 5 (F - 32) 9

B) C = 9 (F - 32) 5

33) C) C =

5 F - 32

D) C = F - 32 9

6

34) A =

1 bh, for b 2

A) b = Ah 2

34) B) b = h 2A

C) b = A 2h

h

6. Basics of Functions and their Graphs: Determine whether the relation is a function. 35) {(-7, -1), (-7, 2), ( -1, 8), (3, 3), (10, -7)} A) Not a function

35) B) Function

Evaluate the function at the given value of the independent variable and simplify. 36) f(x) = -3x - 8; f( -2) A) 22 B) -2 C) 14 37) f(x) = x + 11; f(-2) A) -3 C) 1.73

36) D) -11 37)

B) 3 D) not a real number

Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x. 38)

38)

A) function 39)

39)

B) function

7

40)

40)

B) not a function Use the graph to find the indicated function value. 41) y = f(x). Find f(-1)

A) -0.2

B) -4.2

41)

D) 0.2

8

Use the graph to determine the function's domain and range. 42)

domain: (-∞, ∞) range: [-4, ∞) C) domain: [-1, ∞) range: [-4, ∞)

42)

B) domain: (-∞, ∞) range: (-∞, ∞) D) domain: (-∞, -1) or (-1, ∞) range: (-∞, -4) or (-4, ∞)

43)

43)

A) domain: [0, ∞) range: [0, ∞) domain: [0, ∞) range: [-1, ∞)

B) domain: [0, ∞) range: (-∞, ∞) D) domain: (-∞, ∞) range: [-1, ∞)

9

Identify the intervals where the function is changing as requested. 44) Increasing

A) (-3, 3)

B) (-3, ∞)

44)

C) (-2, ∞)

45) Constant

45)

B) (2, ∞)

C) (-2, -1)

Evaluate the piecewise function at the given value of the independent variable. 46) f(x) = 3x + 1 if x < -1 ; f(2) -2x - 5 if x ≥ -1 A) -8 C) 1

D) (1, 2)

46) D) -3

7. Linear Functions and Slope: Find the slope of the line that goes through the given points. 47) (-2, -6), ( -9, -17) 11 B) 7 7

47) 7 C) 11

Use the given conditions to write an equation for the line in point-slope form. 48) Slope = 4, passing through (-3, 7) A) x - 7 = 4(y + 3) B) y = 4x + 19 C) y + 7 = 4(x - 3) ) y - 7 = 4(x + 3)

23 D) 11

48)

10

Use the given conditions to write an equation for the line in slope-intercept form. 2 49) Slope = , passing through (7, 3) 3 A) y =

2 x+7 3

3

50) Passing through (-8, -2) and ( -5, -7) 5 46 )y=- x3 3 C) y = mx -

46 3

3

C) y = mx -

D) y =

2 + 5 x 3 3 50)

5 B) y + 2 = - (x + 8) 3 D) y =

Graph the line whose equation is given. 51) y = 2x - 2

A)

5 3

49)

5 - 46 x 3 3

51)

B)

C)

11

Determine the slope and the y-intercept of the graph of the equation. 52) 7x - 10y - 70 = 0 10 A) m = ; (0, 10) B) m = 7; (0, 70) 7 m=

7 ; (0, -7) 10

D) m = -

7 ; (0, 7) 10

Use the given conditions to write an equation for the line in the indicated form. 53) Passing through (2, 3) and parallel to the line whose equation is y = -2x + 3 ; point-slope form A) y - 2 = -2(x - 3) C) y = 2x D) y - 3 = x - 2 54) Passing through (5, 3) and perpendicular to the line whose equation is y = 2x + 7; point-slope form 1 A) y - 3 = (x + 5) ) 2 2 C) y - 5 =

1 (x - 3) 2

52)

53)

54)

D) y = - 2 x - 11

12

8. Transformations of Functions: Begin by graphing the standard quadratic function f(x) = x2 . Then use transformations of this graph to graph the given function. 55) h(x) = (x - 7)2 - 5

A)

C)

D)

13

Use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g. 56) g(x) = -f(x - 1) + 2

56)

y = f(x)

A)

B)

C)

9. Combinations of Functions; Composite Functions; Inverse Functions: Given functions f and g, perform the indicated operations. 57) f(x) = 3 - 5 x, g(x) = -8x + 5 Find f + g. A) -5x B) 3x + 8

57) C) -8x + 3

For the given functions f and g , find the indicated composition. 58) f(x) = 3x + 9, g(x) = 5x - 1 (f ∘g)(x) A) 15x + 8 B) 15x + 44

58) D) 15x + 12

14

59) f(x) = x 2 + 2 x + 2, (f ∘g)(-3) A) 51

g(x) = x 2 - 2 x - 3

59)

B) 136

D) 17

Find the inverse of the one-to-one function. 60) f(x) = -4x + 5

60) y-5 B) f-1(x) = -4

C) f-1(x) = x + 5 -4

61) f(x) =

D) f-1(x) = -4x - 5 -4

x-7

61)

A) f-1(x) = x2 - 7

B) f-1(x) = x2 + 7

C) f-1(x) = x + 7

D) f-1(x) =

1 2 x +7

10. Distance and Midpoint Formula. Find the distance between the pair of points. 62) (-1, 4) and ( -5, 7) A) 6

62) C) 10

Find the midpoint of the line segment whose end points are given. 63) (7, 3) and (4, 1) 3 A) ( , 1) B) (11, 4) C) (3, 2) 2

D) 25

63) 2

11. Circles: Write the standard form of the equation of the circle with the given center and radius. 64) (-4, 4); 3 B) (x - 4)2 + (y + 4)2 = 9 C) (x + 4)2 + (y - 4)2 = 3

64)

D) (x - 4)2 + (y + 4)2 = 3

Find the center and the radius of the circle. 65) (x - 5)2 + (y + 7)2 = 36 A) (-7, 5), r = 6

B) (7, -5), r = 36

65) D) (-5, 7), r = 36

15

Graph the equation. 66) (x - 1)2 + (y - 2)2 = 49

66)

A)

B)

Domain = (-6, 8), Range = (-5, 9)

Domain = (-8, 6), Range = (-9, 5)

13. Complex Numbers: Add or subtract as indicated and write the result in standard form. 67) -7 - (- 2 - 7i) - ( - 2 + 5i) A) 4 - 2i B) -3 + 2i C) -3 - 2i

D) 4 + 2i

Find the product and write the result in standard form. 68) (-3 - 7i)(2 + i) A) -13 + 11i B) -13 - 17i

D) 1 - 17i

67)

68) C) 1 + 11i

Divide and express the result in standard form. 8 69) 4+i A)

70)

32 - 8 i 17 17

B)

32 - 8 i 15 15

69) C)

32 + 8 i 15 15

D)

32 + 8 i 17 17

6 - 6i 8 + 2i A) 1 -

70) 1 i 4

B)

60 + 36 i 17 17

C)

3 - 1 i 4 20

D)

9 - 15 i 17 17

16

Perform the indicated operations and write the result in standard form. 71)

72)

-16 + -81 A) -13

71) B) 13i

C) 36i

D) -13i

-24

-2 -

72)

2 A) 1 + i

6

B) -1 - i

6

C) -1 + i

6

D) -1 - i

2

14. Quadratic Functions: Solve the equation by factoring. 73) x2 = x + 6

73) B) {1, 6}

C) {-2, -3}

B) {-12, 1}

C) {12, 10}

D) {2, 3}

Solve the equation by factoring. 74) x2 + 2 x - 120 = 0

74)

A) {12, -10}

Solve the equation by the square root property. 75) 6x2 = 54 A) {-3

6, 3

6}

76) (x - 3)2 = 49 A) {52}

75)

B) {-6, 6}

C) {-3, 3}

D) {0}

B) {-10, -4}

C) {-7, 7}

D) {-4, 10}

76)

Solve the equation using the quadratic formula. 77) x2 + 7x + 7 = 0 A) C)

-7 - 21 -7 + 21 , 14 14 -7 2

77 , -7 +

77) B)

77

D)

2

-7 - 21 -7 + 21 , 2 2 72

21 , 7 +

21 2

78) 5x2 - 3x + 3 = 0 A)

3 ± i 51 10

78) B)

-3 ± 51 10

C)

-3 ± i 51 10

D)

3 ± 51 10

17

The graph of a quadratic function is given. Determine the function's equation. 79)

A) h(x) = (x - 2)2 + 2 C) j(x) = (x - 2)2 - 2

79)

B) g(x) = (x + 2)2 - 2

80)

80)

A) f(x) = -x2 - 2x - 1

B) g(x) = -x2 + 2x + 1 D) h(x) = -x2 - 1

Find the coordinates of the vertex for the parabola defined by the given quadratic function. 81) f(x) = (x - 4)2 - 4 A) (0, -4) 82) y + 4 = (x - 2)2 A) (2, - 4)

B) (4, 4)

82) B) (- 2, - 4)

C) (4, - 2)

D) (4, 2)

Find the axis of symmetry of the parabola defined by the given quadratic function. 83) f(x) = x 2 + 7 A) x = -7 84) f(x) = (x + 4)2 - 6 A) x = 6

81)

D) (-4, 0)

83)

B) x = 7

D) y = 7

B) x = -6

D) x = 4

84)

18

15. Polynomial and Rational Functions Form a polynomial whose zeros and degree are given. 85) Zeros: -3, -2, 2; degree 3 A) f(x) = x3 - 3x2 + 4 x - 12 for a = 1 C) f(x) = x3 - 3x2 - 4 x + 12 for a = 1

85) B) f(x) = x3 + 3x2 + 4 x + 12 for a = 1 D) f(x) = x3 + 3x2 - 4x - 12 for a = 1

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. 86) 86) f(x) = 5(x + 3)(x - 3)3 A) -3, multiplicity 1, crosses x-axis; 3, multiplicity 3, crosses x-axis B) 3, multiplicity 1, touches x-axis; -3, multiplicity 3 C) 3, multiplicity 1, crosses x-axis; -3, multiplicity 3, crosses x-axis D) -3, multiplicity 1, touches x-axis; 3, multiplicity 3 87) f(x) = 2(x2 + 4)(x + 1)2 A) -1, multiplicity 2, touches x-axis B) -1, multiplicity 2, crosses x-axis C) -4, multiplicity 1, crosses x-axis; -1, multiplicity 2, touches x-axis D) -4, multiplicity 1, touches x-axis; -1, multiplicity 2, crosses x-axis Find the x - and y-intercepts of f. 88) f(x) = (x + 4)(x - 2)(x + 2) A) x-intercepts: -4, -2, 2; y -intercept: -16 C) x-intercepts: -4, -2, 2; y -intercept: 16 89) f(x) = 4x - x3 A) x-intercepts: 0, -4; y-intercept: 0 C) x-intercepts: 0, 2, -2; y-intercept: 4

88) B) x-intercepts: -2, 2, 4; y-intercept: -16 D) x-intercepts: -2, 2, 4; y -intercept: 16 89) B) x-intercepts: 0, 2, -2; y-intercept: 0 D) x-intercepts: 0, -4; y-intercept: 4

Find the domain of the rational function. 2x 90) g(x) = x+2 A) {x|x ≠ 0} C) {x|x ≠ -2} 91) f(x) =

92) f(x) =

90) B) all real numbers D) {x|x ≠ 2}

x+7 x2 - 9

A) {x|x ≠ -3, x ≠ 3, x ≠ -7} C) {x|x ≠ -3, x ≠ 3}

91) B) {x|x ≠ 0, x ≠ 9} D) all real numbers

x+2

92)

x2 + 16x

A) {x|x ≠ -4, x ≠ 4} C) {x|x ≠ 0, x ≠ -16}

87)

B) all real numbers D) {x|x ≠ -4, x ≠ 4, x ≠ -2}

19

Find the vertical asymptotes of the rational function. 4x2 93) h(x) = (x + 2)(x - 6) A) x = 2, x = -6 C) x = -2, x = 6 94) g(x) =

93) B) x = -2, x = 6, x = -4 D) x = -4

x+4

94)

x2 + 4

A) x = -2, x = 2 C) x = -2, x = -4

B) none D) x = -2, x = 2, x = -4

List the potential rational zeros of the polynomial function. Do not find the zeros. 95) f(x) = 6x4 + 2x3 - 3x2 + 2 1 1 1 2 A) ± , ± , ± , ± , ± 1, ± 2, ± 3 6 3 2 3 C) ±

1 1 1 2 , ± , ± , ± , ± 1, ± 2 6 3 2 3

95)

1 1 1 B) ± , ± , ± , ± 1, ± 2 6 3 2 D) ±

1 3 , ± , ± 1, ± 2, ± 3, ± 6 2 2

Use the Remainder Theorem to find the remainder when f(x) is divided by x - c. 96) f(x) = x4 + 8x3 + 12x2; x + 1 A) R = 21

B) R = -21

C) R = -5

96) D) R = 5

Form a polynomial f(x) with real coefficients having the given degree and zeros. 97) Degree 3: zeros: 1 + i and -5 A) f(x) = x3 + x 2 - 8 x + 10 B) f(x) = x3 -5x2 - 8 x - 12 C) f(x) = x3 + 3x2 - 8 x + 10 D) f(x) = x3 + 3x2 + 10 x - 8 Use the given zero to find the remaining zeros of the function. 98) f(x) = x 4 - 21x2 - 100; zero: -2i A) 2i, 5i, -5i

B) 2i, 10, -10

97)

98) C) 2i, 10i, -10i

D) 2i, 5, -5

16. Rational Expressions

Perform the indicated operations and simplify the result. Leave the answer in factored form. 4x - 4 ∙ 8x2 99) 5x - 5 x A)

32x 5

B)

20x2 + 40 x + 20 8x3

C)

32x3 - 32x2 5x2 - 5x

D)

5 32x

99)

20

100)

x2 - 10x + 24 x2 - 8x + 15 ∙ x 2 - 4x + 3 x2 - 14x + 48

100)

A)

(x + 4)(x + 5) (x + 1)(x + 8)

B)

(x2 - 10x + 24)(x2 - 8x + 15) (x2 - 4x + 3)(x2 - 14x + 48)

C)

(x - 4)(x - 5) (x - 1)(x - 8)

D)

(x - 4) (x - 8)

Graph the function. 101) f(x) = 2x x-3

101)

A)

B)

C)

D)

21

Find the vertical asymptotes of the rational function. x+4 102) h(x) = x2 - 36 A) x = 36, x = -4 C) x = -6, x = 6

102) B) x = 0, x = 36 D) x = -6, x = 6, x = -4

Give the equation of the horizontal asymptote, if any, of the function. x2 + 5x - 3 103) g(x) = x-3 A) y = 0

B) y = 1

C) y = 3

103) D) none

Graph the function. 104) f (x) = A)

5 3x + 8

104) B)

C)

22

Give the equation of the horizontal asymptote, if any, of the function. 3 105) h(x) = 7x - 5 x - 3 8x + 9 A) y = 0

B) y = 7

C) y =

105) 7 8

Use the graph of the rational function shown to complete the statement. 106)

As x →-2-, f(x)→ ? A) +∞

B) 2

106)...