Review for Midterm #2

REVIEW FOR MIDTERM #2

Multiple Choice Identify the choice that best completes the statement or answers the question.

Determine whether the function is linear or quadratic. Identify the quadratic, linear, and constant terms.

____ 1. a. linear function c. linear function linear term: linear term: constant term: 6 constant term: –6 b. quadratic function d. quadratic function quadratic term: quadratic term: linear term: linear term: constant term: 6 constant term: –6

____ 2. a. linear function c. quadratic function linear term: quadratic term: constant term: –6 linear term: constant term: –6 b. quadratic function d. linear function quadratic term: linear term: linear term: constant term: –6 constant term: –6

Identify the vertex and the axis of symmetry of the parabola. Identify points corresponding to P and Q.

4 Q ____ 3. –8 –4 O 4 8 x P

a. (–1, –2), x = –1 c. (–1, –2), x = –1 P'(0, –1), Q'(–3, 2) P'(–2, –1), Q'(–1, 2) b. (–2, –1), x = –2 d. (–2, –1), x = –2 P'(–2, –1), Q'(–1, 2) P'(0, –1), Q'(–3, 2) y 8

P ____ 4. –8 –4 O 4 8 x Q –4

a. (–3, 1), x = –3; c. (–3, 1), x = –3; P'(–2, 0), Q'(–5, –3) P'(–4, 0), Q'(–1, –3) b. (1, –3), x = 1; d. (1, –3), x = 1; P'(–2, 0), Q'(–5, –3) P'(–4, 0), Q'(–1, –3) ____ 5. Find a quadratic function to model the values in the table. Predict the value of y for x = 6.

x y –1 2 0 –2 3 10

a. ; –58 c. ; 58 b. ; 60 d. ; –58

Find a quadratic model for the set of values.

____ 6. (–2, 8), (0, –4), (4, 68) a. c. b. d.

____ 7. x –2 0 4 f(x) 1 –3 85

a. c. b. d. ____ 8. A biologist took a count of the number of migrating waterfowl at a particular lake, and recounted the lake’s population of waterfowl on each of the next six weeks. Week 0 1 2 3 4 5 6 Population 585 582 629 726 873 1,070 1,317

a. Find a quadratic function that models the data as a function of x, the number of weeks.

b. Use the model to estimate the number of waterfowl at the lake on week 8.

a. ; 1,614 waterfowl b. ; 2,679 waterfowl c. ; 1,961 waterfowl d. ; 2,201 waterfowl

____ 9. A manufacturer determines that the number of drills it can sell is given by the formula , where p is the price of the drills in dollars. a. At what price will the manufacturer sell the maximum number of drills? b. What is the maximum number of drills that can be sold?

a. $60; 285 drills c. $31; 2,418 drills b. $30; 2,415 drills d. $90; 8,385 drills ____ 10. Dalco Manufacturing estimates that its weekly profit, P, in hundreds of dollars, can be approximated by the formula , where x is the number of units produced per week, in thousands. a. How many units should the company produce per week to earn the maximum profit? b. Find the maximum weekly profit.

a. 1,000 units; $1300 c. 1,000 units; $600 b. 3,000 units; $100 d. 2,000 units; $1100 ____ 11. Which is the graph of ? a. y c. y 8 8 6 6 4 4 2 2

–8 –6 –4 –2 O 2 4 6 8 x –8 –6 –4 –2 O 2 4 6 8 x –2 –2 –4 –4 –6 –6 –8 –8

b. y d. y 8 8 6 6 4 4 2 2

–8 –6 –4 –2 O 2 4 6 8 x –8 –6 –4 –2 O 2 4 6 8 x –2 –2 –4 –4 –6 –6 –8 –8

____ 12. Use vertex form to write the equation of the parabola. y 8 6 4 2

–8 –6 –4 –2 O 2 4 6 8 x –2 –4 –6 –8

a. c. b. d. ____ 13. Identify the vertex and the y-intercept of the graph of the function . a. vertex: (–2, 5); c. vertex: (2, 5); y-intercept: –7 y-intercept: –7 b. vertex: (2, –5); d. vertex: (–2, –5); y-intercept: –12 y-intercept: 9

____ 14. Write in vertex form. a. c. b. d.

Write the equation of the parabola in vertex form.

____ 15. vertex (–4, 3), point (4, 131) a. c. b. d.

____ 16. vertex (0, 3), point (–4, –45) a. c. b. d.

Factor the expression.

____ 17. a. c. b. d.

____ 18. a. c. b. d.

____ 19. a. c. b. d.

____ 20. a. c. b. d.

____ 21. a. c. b. d.

____ 22. a. c. b. d.

____ 23. a. c. b. d.

____ 24. a. c. b. d.

____ 25. a. c. b. d.

____ 26. a. c. b. d.

____ 27. a. c. b. d.

____ 28. a. c. b. d. no solution

____ 29. Solve by factoring. = 0 a. 1 b. –8, 4 c. –8, 1 d. 1 8,  1,  2 2

Solve the equation by finding square roots.

____ 30. a. 7 c.

b. 7, – 7 d.

____ 31. a. 49 49 b. 7 7 c. 6 6 d. 36 36  ,  ,  ,  , 36 36 6 6 7 7 49 49

____ 32. The function models the height y in feet of a stone t seconds after it is dropped from the edge of a vertical cliff. How long will it take the stone to hit the ground? Round to the nearest hundredth of a second. a. 7.79 seconds c. 0.25 seconds b. 11.02 seconds d. 5.51 seconds

____ 33. Use a graphing calculator to solve the equation . If necessary, round to the nearest hundredth. a. 1.16, –1.16 c. 2.95, –1.7 b. 1.47, –0.85 d. 0.85, –1.47 ____ 34. A landscaper is designing a flower garden in the shape of a trapezoid. She wants the length of the shorter base to be 3 yards greater than the height, and the length of the longer base to be 5 yards greater than the height. For what height will the garden have an area of 360 square yards? Round to the nearest tenth of a yard. a. 17.1 yards c. 39.2 yards b. 34.2 yards d. 152.6 yards

____ 35. Simplify using the imaginary number i. a. b. c. d.

Write the number in the form a + bi.

____ 36. a. c. b. d.

____ 37. –6 – a. c. b. d.

____ 38. Find . a. –9 b. 9 c. 41 d.

____ 39. Identify the graph of the complex number . a. Imaginary Axis c. Imaginary Axis

Real Axis Real Axis

–4 –2 O 2 4 –4 –2 O 2 4

–2 –2

–4 –4

b. Imaginary Axis d. Real Axis

Real Axis Imaginary Axis –4 –2 O 2 4 –4 –2 O 2 4

–2 –2

–4 –4

____ 40. Find the additive inverse of . a. c. b. d.

Simplify the expression.

____ 41. a. c. b. d.

____ 42. a. c. b. d.

____ 43. a. 36 b. –36 c. –36i d. 36i

____ 44. a. c. b. d.

Solve the equation. ____ 45. a. 4 4 c. 3 3  i, i  i, i 3 3 4 4 b. 16 16 d. 4 4  i, i  , 9 9 3 3

____ 46. a. 14 b. –8 c. 4 d. –6

____ 47. a. 1 b. –7 c. –1 d. –1

____ 48. a. –5, 11 b. 5 c. 11 d. –11

____ 49. Find the first three output values of the fractal-generating function . Use z = 0 as the first input value. a. , 536828 + 336604i b. , 536828 + 336604i c. d. , 536828 + 336604i

____ 50. Two complex numbers a + bi and c + di are equal when a = c and b = d. Solve the equation for x and y, where x and y are real numbers. a. 1 2 c. 3 x = ; y =  x = 3; y =  3 3 2 b. 1 2 d. 3 x =  ; y = x = 3; y = 3 3 2

Use the Quadratic Formula to solve the equation.

____ 51. a. 2 b. 1 c. 56 d. 1 , 4 , 2 , 13 2,  5 5 5 5

____ 52. a. 1 c. 1

4 2 b. d. 1 4 4

____ 53. a. 1 c. 1

8 8 b. d. 1 8 4

____ 54. A landscaper is designing a flower garden in the shape of a trapezoid. She wants the shorter base to be 3 yards greater than the height and the longer base to be 7 yards greater than the height. She wants the area to be 155 square yards. The situation is modeled by the equation . Use the Quadratic Formula to find the height that will give the desired area. Round to the nearest hundredth of a yard. a. 12.7 yards c. 10.2 yards b. 20.4 yards d. 320 yards ____ 55. Classify –3x5 – 2x3 by degree and by number of terms. a. quintic binomial c. quintic trinomial b. quartic binomial d. quartic trinomial ____ 56. Classify –7x5 – 6x4 + 4x3 by degree and by number of terms. a. quartic trinomial c. cubic binomial b. quintic trinomial d. quadratic binomial ____ 57. Zach wrote the formula w(w – 1)(5w + 4) for the volume of a rectangular prism he is designing, with width w, which is always has a positive value greater than 1. Find the product and then classify this polynomial by degree and by number of terms. a. ; quintic trinomial b. ; quadratic monomial c. ; cubic trinomial d. ; quartic trinomial

____ 58. Write the polynomial in standard form. a. c. b. d. ____ 59. Write 4x2(–2x2 + 5x3) in standard form. Then classify it by degree and number of terms. a. 2x + 9x4; quintic binomial c. 2x5 – 8x4; quintic trinomial b. 20x5 – 8x4; quintic binomial d. 20x5 – 10x4; quartic binomial ____ 60. Use a graphing calculator to determine which type of model best fits the values in the table.

x –6 –2 0 2 6 y –6 –2 0 2 6

a. quadratic model c. linear model b. cubic model d. none of these

____ 61. Use a graphing calculator to find a polynomial function to model the data. x 1 2 3 4 5 6 7 8 9 10 f(x) 12 4 5 13 9 16 19 16 24 43

a. f(x) = 0.8x4 – 1.73x3 + 12.67x2 – 34.68x + 35.58 b. f(x) = 0.08x3 – 1.73x2 + 12.67x + 35.58 c. f(x) = 0.08x4 + 1.73x3 – 12.67x2 + 34.68x – 35.58 d. f(x) = 0.08x4 – 1.73x3 + 12.67x2 – 34.68x + 35.58 ____ 62. The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 1995. Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in 2006. Years since 1995 1 3 5 7 9 Trees planted (in thousands) 1.3 18.3 70.5 177.1 357.3

a. ; 630.3 thousand trees b. ; 630.3 thousand trees c. ; 618.1 thousand trees d. ; 618.1 thousand trees

____ 63. The table shows the number of llamas born on llama ranches worldwide since 1988. Find a cubic function to model the data and use it to estimate the number of births in 1999.

Years since 1988 1 3 5 7 9 Llamas born (in thousands) 1.6 20 79.2 203.2 416

a. ; 741,600 llamas b. ; 563,200 llamas c. ; 741,600 llamas d. ; 563,200 llamas

____ 64. Write the expression (x + 6)(x – 4) as a polynomial in standard form. a. x2 – 10x + 2 c. x2 + 2x – 24 b. x2 + 10x – 24 d. x2 + 10x – 10 ____ 65. Write 4x3 + 8x2 – 96x in factored form. a. 6x(x + 4)(x – 4) c. 4x(x + 6)(x + 4) b. 4x(x – 4)(x + 6) d. –4x(x + 6)(x + 4) ____ 66. Miguel is designing shipping boxes that are rectangular prisms. One shape of box with height h in feet, has a volume defined by the function . Graph the function. What is the maximum volume for the domain ? Round to the nearest cubic foot. a. 10 ft3 b. 107 ft3 c. 105 ft3 d. 110 ft3

____ 67. Use a graphing calculator to find the relative minimum, relative maximum, and zeros of . If necessary, round to the nearest hundredth. a. relative minimum: (–62.24, 0.36), relative maximum: (37.79, –3.69), zeros: x = 5, –2, 2 b. relative minimum: (0.36, –62.24), relative maximum: (–3.69, 37.79), zeros: x = –5, –2, 2 c. relative minimum: (0.36, –62.24), relative maximum: (–3.69, 37.79), zeros: x = 5, –2 d. relative minimum: (–62.24, 0.36), relative maximum: (37.79, –3.69), zeros: x = –5, –2

____ 68. Find the zeros of . Then graph the equation. a. 3, 2, –3 c. 3, 2 y y

–6 –4 –2 2 4 6 x –6 –4 –2 2 4 6 x –2 –2

–4 –4

–6 –6

b. 0, –3, –2 d. 0, 3, 2 y y

–6 –4 –2 2 4 6 x –6 –4 –2 2 4 6 x –2 –2

–4 –4

–6 –6

____ 69. Write a polynomial function in standard form with zeros at 5, –4, and 1. a. c. b. d.

____ 70. Find the zeros of and state the multiplicity. a. 2, multiplicity –3; 5, multiplicity 6 b. –3, multiplicity 2; 6, multiplicity 5 c. –3, multiplicity 2; 5, multiplicity 6 d. 2, multiplicity –3; 6, multiplicity 5 ____ 71. Divide by x + 3. a. c. b. , R –93 d. , R 99 ____ 72. Determine which binomial is not a factor of . a. x + 4 c. x – 5 b. x + 3 d. 4x + 3 ____ 73. Determine which binomial is a factor of . a. x + 5 b. x + 20 c. x – 24 d. x – 5 ____ 74. The volume of a shipping box in cubic feet can be expressed as the polynomial . Each dimension of the box can be expressed as a linear expression with integer coefficients. Which expression could represent one of the three dimensions of the box? a. x + 6 c. 2x + 3 b. x + 1 d. 2x + 1

Divide using synthetic division.

____ 75. a. c. b. d.

____ 76. a. , R 70 c. , R 46 b. , R –62 d. , R –38

____ 77. Use synthetic division to find P(2) for . a. 2 b. 28 c. 4 d. –16

Solve the equation by graphing.

____ 78. a. x = 49 b. no solution c. x = 19 d. x = 12 ____ 79. a. no solution c. 0, 2, –0.38 b. –2, 0.38 d. 0, –2, 0.38 ____ 80. a. 3 b. –3 c. –3, 3 d. no solution ____ 81. The dimensions in inches of a shipping box at We Ship 4 You can be expressed as width x, length x + 5, and height 3x – 1. The volume is about 7.6 ft3. Find the dimensions of the box in inches. Round to the nearest inch. a. 15 in. by 20 in. by 44 in. c. 15 in. by 20 in. by 45 in. b. 12 in. by 17 in. by 35 in. d. 12 in. by 17 in. by 36 in. ____ 82. Over two summers, Ray saved $1000 and $600. The polynomial represents her savings after three years, where x is the growth factor. (The interest rate r is x – 1.) What is the interest rate she needs to save $1850 after three years? a. 9.3% b. 1.1% c. –269.3% d. 0.1% ____ 83. Solve . Find all complex roots. a. 7 c. 7  , , 5 5 b. no solution d. 7 7  , 5 5

____ 84. Ian designed a child’s tent in the shape of a cube. The volume of the tent in cubic feet can be modeled by the equation , where s is the side length. What is the side length of the tent? a. 4 feet b. 16 feet c. 64 feet d. 8 feet ____ 85. Solve . a. no solution c. 3, –3, 5, –5 b. 3, –5 d. 3, –3 ____ 86. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation . Do not find the actual roots. a. –4, –2, –1, 1, 2, 4 c. 1, 2, 4 b. no roots d. –4, –1, 1, 4 ____ 87. Find the rational roots of . a. 2, 6 b. –6, –2 c. –2, 6 d. –6, 2

Find the roots of the polynomial equation.

____ 88. a. –3 ± 5i, –4 c. –3 ± i, 4 b. 3 ± 5i, –4 d. 3 ± i, 4 ____ 89. a. c.

____ 90. a. c. b. d.

____ 91. A polynomial equation with rational coefficients has the roots . Find two additional roots. a. c. b. d.

____ 92. Find a third-degree polynomial equation with rational coefficients that has roots –5 and 6 + i. a. c. b. d. ____ 93. Find a quadratic equation with roots –1 + 4i and –1 – 4i. a. c. b. d. ____ 94. For the equation , find the number of complex roots and the possible number of real roots. a. 4 complex roots; 0, 2 or 4 real roots b. 4 complex roots; 1 or 3 real roots c. 3 complex roots; 1 or 3 real roots d. 3 complex roots; 0, 2 or 4 real roots

For the equation, find the number of complex roots, the possible number of real roots, and the possible rational roots.

____ 95. a. 7 complex roots; 1, 3, 5, or 7 real roots; possible rational roots: ±1, ±5 b. 7 complex roots; 2, 4, or 6 real roots; possible rational roots: ±1, ±5 c. 5 complex roots; 1, 3, or 5 real roots; possible rational roots: , ±1, ±5 d. 5 complex roots; 1, 3, or 5 real roots; possible rational roots: ±1, ±5 ____ 96. a. 6 complex roots; 2, 4, or 6 real roots; possible rational roots: b. 6 complex roots; 2, 4, or 6 real roots; possible rational roots: c. 6 complex roots; 0, 2, 4, or 6 real roots; possible rational roots: d. 6 complex roots; 0, 2, 4, or 6 real roots; possible rational roots:

____ 97. Find all zeros of . a. c.

Use Pascal’s Triangle to expand the binomial.

____ 98. a. b. c. d.

____ 99. a. b. c. d.

____ 100. a. b. c. d. ____ 101. A manufacturer of shipping boxes has a box shaped like a cube. The side length is 5a + 4b. What is the volume of the box in terms of a and b? a. c. b. d.

____ 102. Use the Binomial Theorem to expand . a. b. c. d. ____ 103. Determine the probability of getting four heads when tossing a coin four times. a. 0.5 b. 0.375 c. 0.25 d. 0.0625 ____ 104. Determine the probability that you will get 3 green lights in a series of 5 lights. Assume red and green are equally likely occurrences. a. 31.25% b. 62.5% c. 10% d. 31.25% ____ 105. Find all the real square roots of 0.0004. a. 0.00632 and –0.00632 c. 0.0002 and –0.0002 b. 0.06325 and –0.06325 d. 0.02 and –0.02

____ 106. Find all the real square roots of . a. no real root c. and b. d.

____ 107. Find all the real cube roots of 0.000027. a. 0.03 c. 0.0009 and –0.0009 b. 0.0009 d. 0.03 and –0.03

____ 108. Find all the real fourth roots of . a. c. and , , , and b. d. and

Find the real-number root.

____ 109. a. 1.3 c. 0.85 b. 2.86 d. no real number root

____ 110. a. –1.6 c. –1.28 b. 1.6 d. no real number root ____ 111.

a. b. c. d.

Simplify the radical expression. Use absolute value symbols if needed.

____ 112. a. b. c. d.

____ 113. a. b. c. d.

____ 114. The formula for the volume of a sphere is . Find the radius, to the nearest hundredth, of a sphere with a volume of 15 in.3. a. 3.58 in. b. 258.01 in. c. 1.53 in. d. 1.85 in.

Multiply and simplify if possible.

____ 115. a. b. c. d. not possible

____ 116. a. –3 b. 3 c. d. not possible

____ 117. a. c. b. d.

____ 118. Simplify . Assume that all variables are positive. a. c. b. d. none of these

____ 119. Multiply and simplify . Assume that all variables are positive. a. c. b. d. none of these

Divide and simplify. ____ 120.

a. b. c. d.

____ 121.

a. b. c. d.

Divide and simplify. Assume that all variables are positive.

____ 122.

a. b. c. d.

____ 123.

a. c. b. d. none of these

Rationalize the denominator of the expression. Assume that all variables are positive.

____ 124.

a. b. c. d. none of these

____ 125.

b. d. none of these

____ 126.

b. d. ____ 127.

Add if possible.

____ 128. a. c. b. d. not possible to simplify

Subtract if possible.

____ 131. a. c. 1 b. d. not possible to simplify

____ 132. A garden has width and length . What is the perimeter of the garden in simplest radical form? a. units c. 91 units b. units d. units

Simplify.

____ 133. a. c. b. d. none of these

____ 134. a. b. c. 20 d. 1

____ 135. a. 9 b. c. 3 d.

____ 136. a. 512 b. 4,096 c. 16 d.

____ 137. a. b. –22.5 c. –3.6 d.

Multiply.

____ 138. a. c. b. d.

____ 139. a. c. b. d.

____ 140. a. c. b. d.

____ 141. a. b. –3 c. 17 d.

____ 142. A rope is units long. The rope is cut into two pieces, so that the lengths of the pieces are in the ratio 2 : 3. What is the length of the longer piece expressed in simplest radical form? a. units c. units b. units d. units

____ 143. Write the exponential expression in radical form. a. b. c. d.

____ 144. Write the radical expression in exponential form. a. b. c. d.

____ 145. Write in simplest form. a. c.

b. d. none of these ____ 146. The area of a circular trampoline is 112.07 square feet. What is the radius of the trampoline? Round to the nearest hundredth. a. 3.37 feet c. 5.97 feet b. 10.59 feet d. 35.67 feet

Solve. Check for extraneous solutions.

____ 147. a. 2 b. 1 c. 2 d. 1  1 and  3 3

____ 148. a. 7 b. 2 c. 1 d. 6  6 3 4 7

____ 149. Let and . Find f(x) + g(x). a. 2x – 4 b. –8x – 8 c. –8x – 4 d. 2x – 8

____ 150. Let and . Find f(x) – g(x). a. 2x – 5 b. 2x + 5 c. 4x – 1 d. 2x – 1

____ 151. Let and . Find 2f(x) – 3g(x). a. c. b. d.

____ 152. Let and . Find and its domain. a. 2 ; all real numbers except x   3 b. ; all real numbers c. ; all real numbers d. 6 ; all real numbers except x   7

____ 153. Let and . Find and its domain.

a. 3; all real numbers b. 3; all real numbers except x  2 c. 1; all real numbers d. –3; all real numbers except x  3

____ 154. Let and . Find and its domain.

a. 3x + 2; all real numbers except x  4 b. –9x + 6; all real numbers except x  4 c. –3x + 2; all real numbers except x  –4 d. 9x – 6; all real numbers except x  –4

____ 155. Let and . Find . a. 23 b. –53 c. –9 d. 3

____ 156. Let and . Find . a. b. c. d.

____ 157. Let and . Find f(g(x)) and g(f(x)). a. f(g(x)) = 10x – 1; g(f(x)) = 10x + 7 b. f(g(x)) = 7x + 3; g(f(x)) = 10x + 7 c. f(g(x)) = –7x – 3; g(f(x)) = –10x + 7 d. f(g(x)) = –10x – 7; g(f(x)) = 7x + 3

Short Answer

158. In an experiment, a petri dish with a colony of bacteria is exposed to cold temperatures and then warmed again. a. Find a quadratic model for the data in the table. b. Use the model to estimate the population of bacteria at 9 hours.

Time (hours) 0 1 2 3 4 5 6 Population (1000s) 5.1 3.03 1.72 1.17 1.38 2.35 4.08

159. The table shows the number of copies of a book sold per 100,000 people in the United States for five selected years. The values in the first column are years since 1987, so corresponds to 1987, corresponds to 1990, and so on. Years since Copies sold per 100,000 1987 (x) people (y) 0 8.3 3 9.4 6 9.5 9 7.4 12 5.7

a. Use a graphing calculator to model the data with a quadratic function. Round the coefficients and constant term to four decimal places. b. Graph the data and the quadratic function.

c. Use the graph or the equation to estimate the number of copies sold per 100,000 people in 1998. d. Would you use the quadratic function to predict the number of copies sold per 100,000 people in 2005? Explain.

160. Graph .

161. Graph . Identify the vertex and the axis of symmetry.

162. Graph . What is the minimum value of the function?

163. Graph . Does the function have a maximum or minimum value? What is this value?

164. A science museum is going to put an outdoor restaurant along one wall of the museum. The restaurant space will be rectangular. Assume the museum would prefer to maximize the area for the restaurant. a. Suppose there is 120 feet of fencing available for the three sides that require fencing. How long will the longest side of the restaurant be? b. What is the maximum area?

165. Graph .

166. In a baseball game, an outfielder throws a ball to the second baseman. The path of the ball is modeled by the

equation , where y is the height of the ball in feet after the ball has traveled x

feet horizontally. The second baseman catches the ball at the same height as the height at which the outfielder released it. a. What was the maximum height of the ball along its path? Answer to the nearest foot. b. How far was the second baseman from the outfielder at the time he caught the ball? c. How high above the ground was the ball when it left the hand of the outfielder?

167. Use the graph of . a. If you translate the parabola to the right 2 units and down 7 units, what is the equation of the new parabola in vertex form? b. If you translate the original parabola to the left 2 units and up 7 units, what is the equation of the new parabola in vertex form? c. How could you translate the new parabola in part (a) to get the new parabola in part (b)?

168. Suppose you cut a small square from a square of fabric as shown in the diagram. Write an expression for the remaining shaded area. Factor the expression. 3

169. For how many integer values of a can be factored? What are they?

170. The Sears Tower in Chicago is 1454 feet tall. The function models the height y in feet of an object t seconds after it is dropped from the top of the building. a. After how many seconds will the object hit the ground? Round your answer to the nearest tenth of a second. b. What is the height of the object 5 seconds after it is dropped from the top of the Sears Tower?

171. A carpenter is cutting a board to make a brace that will go at the bottom of a storage shed wall. The brace will be in the shape of a right triangle. The hypotenuse will be 41 inches long. The longer leg will be 31 inches longer than the shorter leg. a. Let x be the length of the shorter leg. Write a quadratic equation that models the situation. b. Use factoring to solve the equation you wrote in part (a). What are the solutions? c. What is the length of the longer leg of the brace? 172. Two boats leave from the same point at the same time. One boat travels due east and the other travels due north. One boat travels 6 kilometers faster than the other. After 4 hours, the boats are 67 kilometers apart. a. Let x be the speed of the slower boat. Write a quadratic equation that models the situation. b. Use a graphing calculator to solve the equation in part (a) graphically. What are the solutions, to the nearest hundredth? c. What are the speeds of the boats? Round your answers to the nearest hundredth.

173. Determine the type and number of solutions of .

174. A park planner has sketched a rectangular park in the first quadrant of a coordinate grid. Two sides of the park lie on the x- and y-axes. A trapezoidal flower bed will be bounded by the line , the x-axis, and the vertical lines and , where . The area A of the trapezoid is modeled by . Assume that lengths along the axes are measured in meters. For what value of a will the trapezoid have an area of 25 square meters? Use the Quadratic Formula to find the answer.

175. During a manufacturing process, a metal part in a machine is exposed to varying temperature conditions. The manufacturer of the machine recommends that the temperature of the machine part remain below 135F. The temperature T in degrees Fahrenheit x minutes after the machine is put into operation is modeled by . a. Tell whether the temperature of the part will ever reach or exceed 135F. Use the discriminant of a quadratic equation to decide. b. If the machine is in operation for 90 minutes before being turned off, how many times will the temperature of the part be 134F?

176. The diagram shows a storage building that consists of a cubic base and a pyramid-shaped top. a. Write an expression for the cube’s volume. b. Write an expression for the volume of the pyramid-shaped top. c. Write a polynomial expression to represent the total volume.

177. The table shows the population of Rockerville over a twenty-five year period. Let 0 represent 1975.

Population of Rockerville Year Population 1975 336 1980 350 1985 359 1990 366 1995 373 2000 395

a. Find a quadratic model for the data. b. Find a cubic model for the data. c. Graph each model. Compare the quadratic model and cubic model to determine which is a better fit.

178. The volume in cubic feet of a box can be expressed as , or as the product of three linear factors with integer coefficients. The width of the box is 2 – x. a. Factor the polynomial to find linear expressions for the height and the width. b. Graph the function. Find the x-intercepts. What do they represent? c. Describe a realistic domain for the function. d. Find the maximum volume of the box.

179. The volume in cubic feet of a workshop’s storage chest can be expressed as the product of its three dimensions: . The depth is x + 1. a. Find linear expressions with integer coefficients for the other dimensions. b. If the depth of the chest is 6 feet, what are the other dimensions?

180. To raise money, a club is selling 500 raffle tickets. Each ticket has a 5% chance of winning a prize. K’Lynn buys 6 tickets. To the nearest percent, find the probability of each outcome. a. winning exactly 1 time b. winning exactly 2 times c. winning exactly 4 times

181. The optimal height h of the letters of a message printed on pavement is given by the formula . Here d is the distance of the driver from the letters and e is the height of the driver’s eye above the pavement. All of the distances are in meters. Find h for d = 90 m and e = 1.4 m. Show your work.

182. The velocity of sound in air is given by the equation , where v is the velocity in meters per second and t is the temperature in degrees Celsius. a. Find the temperature when the velocity of sound in air is 318 meters per second. Round the answer to the nearest degree. b. Find the velocity of sound in meters per second when the temperature is 20°C. Round the answer to the nearest meter per second.

183. An airplane travels at a constant speed of 375 miles per hour in still air. During a particular portion of the flight, the wind speed is 40 miles per hour in the same direction the plane is flying. a. Write a function f(x) for the distance traveled by the airplane in still air for x hours. b. Write a function g(x) for the effect of the wind on the airplane for x hours. c. Write an expression for the total speed of the airplane flying with the wind.

184. Let and . a. Find f(g(x)). b. Find g(f(x)).

185. Maribel is going to build a rectangular pen for her two dogs. She has 180 feet of fencing. To keep the dogs separate, she plans to put fencing down the middle of the pen to split the large rectangle into two smaller rectangles. What are the dimensions and area of the largest pen area she can use to accommodate both dogs? Show and explain your work.

186. Show that is equal to . Then use this to explain how you know that 5 is the minimum value of the function.

187. Use a graphing calculator to graph the function . a. What does the graph let you conclude about real number solutions of ? Explain. b. Substitute for x in the equation . Simplify. Is the resulting equation true? Show your work. c. What conclusions can you state about solutions of ? Explain.

188. A model for the height of a toy rocket shot from a platform is , where x is the time in seconds and y is the height in feet. a. Graph the function. b. Find the zeros of the function. c. What do the zeros represent? Are they realistic? d. About how high does the rocket fly before hitting the ground? Explain.

189. Find the rational roots of . Explain the process you use and show your work.

190. Use the first 13 rows of Pascal’s triangle. a. Circle the numbers that are multiples of 3. Notice that the multiples of 3 form triangular groups. How many multiples of 3 are in the uppermost group? b. Color the numbers that are multiples of 5. How many multiples of 5 are in the uppermost triangular group? c. Draw a triangle around the numbers that are multiples of 7. How many multiples of 7 are in the triangular group? d. Look for a pattern. Use the pattern to predict the number of multiples of 11 that would form the first triangular group. e. Does your pattern work for multiples of 4 or 9? If not, for what kind of numbers does your pattern seem to work?

191. The formula for the surface area of a sphere is , where A is surface area and r is the radius of the sphere. a. Use the formula to express r in terms of A and rationalize the denominator. Explain your work. b. Suppose the surface area of a child’s ball is about 113 square inches. Find the radius of the ball and explain your steps.

192. Consider the equation . a. Solve the equation and check for extraneous roots. Explain your steps. b. Explain why you must check for extraneous roots when finding the solution or solutions of a radical equation.

193. Spheres are being packed into a square box as shown in the diagram. a. Express the radius r of each sphere as a function of the length x of the sides of the square. b. Express the volume V of a sphere as a function of the radius r. c. Find . d. Find and interpret .

194. A baseball player hits a fly ball that is caught about 4 seconds later by an outfielder. The path of the ball is a parabola. The ball is at its highest point as it passes the second baseman, who is 127 feet from home plate. About how far from home plate is the outfielder at the moment he catches the ball? Explain your reasoning.

195. A data processing consultant charges clients by the hour. His weekly earnings E are modeled by the function , where x is his hourly rate in dollars. Can he earn $2500 in a single week? Explain.

196. Consider system of equations.

Suppose the two parabolas have the same axis of symmetry. How many possible solutions does the system have? Explain.

197. What are multiple zeros? Explain how you can tell if a function has multiple zeros.

198. Use division to prove that x = 3 is a real zero of .

199. A polynomial equation with rational coefficients has the roots and . Explain how to find two additional roots and name them.

200. Rationalize the denominator for the expression . Explain your steps.

201. Michael is shopping for a new CD player with a built-in alarm clock. Electronics City has a special coupon for $30 off any CD player and is also having a sale with a 25% discount on any alarm clock. a. Write a function rule to model a 25%-off sale, and a function rule to model a $30-off coupon. b. Use composition of functions to model how much Michael would pay for a CD alarm clock if the clerk applies the discount first and then the coupon. c. Use composition of functions to model how much Michael would pay for a CD alarm clock if the clerk applies the coupon first and then the discount. d. Michael selects a CD alarm clock with a regular price of $150. How much more will the item cost if the clerk applies the coupon first? e. Why does the CD alarm clock cost less if the discount is applied after the coupon? REVIEW FOR MIDTERM #2 Answer Section

MULTIPLE CHOICE

1. C 2. B 3. A 4. A 5. C 6. A 7. C 8. C 9. B 10. A 11. A 12. C 13. A 14. D 15. A 16. B 17. C 18. D 19. D 20. B 21. A 22. D 23. C 24. C 25. B 26. B 27. B 28. C 29. C 30. B 31. B 32. D 33. B 34. A 35. B 36. C 37. B 38. C 39. B 40. C 41. C 42. C 43. B 44. A 45. A 46. D 47. C 48. A 49. C 50. D 51. B 52. A 53. C 54. C 55. A 56. B 57. C 58. B 59. B 60. C 61. D 62. B 63. C 64. C 65. B 66. C 67. B 68. D 69. B 70. C 71. B 72. A 73. D 74. D 75. D 76. D 77. C 78. B 79. D 80. A 81. A 82. A 83. A 84. A 85. C 86. A 87. B 88. B 89. A 90. D 91. B 92. A 93. C 94. A 95. A 96. D 97. B 98. C 99. D 100. A 101. B 102. D 103. D 104. D 105. D 106. A 107. A 108. B 109. A 110. D 111. D 112. C 113. A 114. C 115. A 116. D 117. A 118. A 119. A 120. A 121. B 122. B 123. A 124. A 125. A 126. C 127. D 128. C 129. D 130. C 131. D 132. B 133. A 134. C 135. C 136. C 137. A 138. B 139. A 140. B 141. B 142. B 143. A 144. C 145. A 146. C 147. D 148. A 149. A 150. B 151. C 152. C 153. B 154. D 155. B 156. D 157. A

SHORT ANSWER

158. a. b. 13,830 bacteria 159. a. b. 12

11 e l

p 10 o e

0 8 0 , 0

d l 5 o S

s 4 e i p

o 3 C 2

1 2 3 4 5 6 7 8 9 10 11 12 years c. about 6.4 copies sold per 100,000 people d. No; the value of y for x = 18 is about , and a negative value for y does not make sense. y 8 6 4 2

–8 –6 –4 –2 O 2 4 6 8 x –2 –4 –6 –8 160. y 8 6 4 2

–8 –6 –4 –2 O 2 4 6 8 x –2 –4 –6 –8 161. vertex: , axis of symmetry:

y 8 6 4 2

–8 –6 –4 –2 O 2 4 6 8 x –2 –4 –6 –8 162. minimum: 1 y 8 6 4 2

–8 –6 –4 –2 O 2 4 6 8 x –2 –4 –6 –8 163. maximum value; 8 164. a. 40 ft b. 1,600 y

–8 –6 –4 –2 O 2 4 6 8 x –2

165. 166. a. 44 ft b. 375 ft c. 5 ft 167. a. b. c. left 4 units, up 14 units 168. 169. 4 integer values; 170. a. 9.5 seconds b. 1,054 ft 171. a. b. c. 40 inches 172. a. b. c. 8.46 km/h, 14.46 km/h 173. two real solutions 174. meters, or about 3.68 meters 175. a. yes b. two times 176. a. b. or

c. or 177. a. b.

The cubic model is a better fit. 178. a. y

–4 4 8 x –4

–8 x-intercepts: x = 0, 2, 4. These are the values of x that produce a volume of 0. c. 0 < x < 2 d. 3.08 cubic feet 179. a. height, x – 1; width, x – 3 b. height, 4 ft; width, 2 ft 180. a. 4% b. 2% c. 0% 181. [4] Write the formula.

Substitute for d and e. =  49.14 The distance is about 49.14 meters. 182. a. –20°C b. 342 meters per second 183. a. f(x) = 375x b. g(x) = 40x c. f(x) + g(x) = 375x + 40x or 415x 184. a. b.

185. [4] Let x be the length of the divider. This will also be the length of two sides of the large rectangle. The length of each of the other two sides of the large rectangle will be half of what is left over when you subtract 3x from the total 180 feet of fencing. The area A of

the large rectangle is the product of the length and width, so .

Simplify the right side to get . For this quadratic function, and .The graph will open down because a is negative. The function will have a maximum value for .

Substitute 30 for x in to find the length of the other side of the large pen.

The largest pen is 30 ft by 45 ft. The area is , or 1350 .

[3] The quadratic model is correct, and the final dimensions and area are correct, but the explanation is incomplete. [2] There is clear evidence that the student understands how to find the maximum value of a quadratic function, but the quadratic model was not correct or the explanation was too sketchy. [1] The model or final dimensions was incorrect and there was little or no work shown. 186. [4]

When , the value of is or 5. If x is any number other than 3, then and is a positive number. So will also be a positive number. The sum of a positive number and 5 has to be greater than 5. Therefore, the value of is 5 when x is 3 and greater than 5 when x is any number other than 3. Since is equal to , the minimum value of the function is 5. [3] Most of the reasoning is correct, but one or two points of the argument were not addressed thoroughly. [2] The reasoning was based too much on specific numerical values of x and y. [1] There were some correct observations, but there was no overall grasp of the situation. 187. [4] a. The equation has no real number solutions. The graph of does not intersect the x-axis. b.

Yes, is true. c. has no real number solutions. But the equation does seem to have as a solution, because you get a true equation when you substitute for x and simplify. [3] Most work is correct, but the student does not see that can be considered to be a solution of the quadratic equation. [2] The student has mistakes in the calculations in part (b), and these prevent correct conclusions in parts (b) and (c). [1] The student gives little evidence of understanding how to use graphs to investigate real number solutions. The calculations with complex numbers are not accurate. 188. [4] a. y

–2 2 4 6 8 10 x –100

–200

b. x –0.05, x 9.11 c. The zeros represent the times at which the height of the rocket is 0. The time – 0.05 seconds is not realistic. The time 9.11 seconds is the time at which the rocket lands. d. about 336 feet; The height is the maximum value of the function. [3] an error in one of the three parts of the question [2] an error in two parts of the question [1] one part missing and errors in answer or reasoning for one of the other parts 189. [4] Step 1: List the possible rational roots by using the Rational Root Theorem. The leading coefficient is 4 with factors of ±1, ±2, and ±4. The constant term is –1 with factors of – 1 and 1. The only possible roots of the equation have the form . Those

roots would be ±1, ± , and ± . Step 2: Test each possible rational root in the equation. The only roots that satisfy the equation are and 1. [3] an error in computation or missing part of the explanation [2] several errors in computation or in the explanation [1] one root given with no explanation 190. [4] a. 3 b. 10 c. 21 d. Answers may vary. Sample: If you subtract 1 from the number, divide the results by 2, then multiply this number by the original number, then you will find how many times the multiple of that number occurs in the uppermost triangular group.

e. No. It works for prime numbers. [3] one incorrect computation or an error in reasoning [2] two computational errors or poor explanation [1] one or more answers missing and/or no explanation 191. [4] a. Begin by isolating on one side of the equation.

Next, take the square root of each side.

Finally, rationalize the denominator by multiplying the numerator and denominator by .

b. Use the formula from part (a), substituting 113 for A and solving for r. The radius is about 3 inches. [3] one error in computation or incomplete explanation [2] two errors in computation or no explanation [1] answer given with no explanation or work shown 192. [4] a. Write the equation.

Square each side.

Simplify.

Combine like terms.

Factor.

Factor Theorem x = 11 or x = 22 Check each root to make sure it is a solution of the equation.

5  –5 6 = 6 b. When you raise both sides of an equation to a power, extraneous roots can be introduced, so you must check that the roots satisfy the original equation. In part a, one of the roots did not satisfy the equation. [3] correct procedure with one mathematical error [2] correct procedure with two mathematical errors [1] incorrect or incomplete procedure OR correct answer with no explanation or work shown 193. [4] a.

c. d. This is the volume of one sphere when the length of each side of the box is 6. [3] only three parts correct [2] only two parts correct [1] only one parts correct

OTHER 194. The outfielder is about 254 feet from home plate. The parabola is symmetric about its axis, which is a vertical line. The second baseman is at a point on this axis of symmetry. So the outfielder is the same distance from the second baseman as the second baseman is from home plate. 127 + 127 = 254, so the distance from home plate to the outfielder is about 254 feet. 195. No; the discriminant of –0.2x2 + 40x – 2500 = 0 is less than 0, so there are no real solutions. 196. 0, 1, 2 or infinitely many solutions

Explanations may vary. Sample: 0: If a and n have opposite signs, and the vertex of the upward-opening parabola is above the downward-opening parabola, there is no point of intersection. This can also be true when a and n have the same sign.

–4 –2 O 2 4 x –4 –2 O 2 4 x –2 –2

–4 –4

1: If the parabolas have the same vertex but , their graphs will only intersect at one point, the vertex.

–4 –2 O 2 4 x –4 –2 O 2 4 x –2 –2

–4 –4

2: If signs of a and n are different and the vertex of the upward-opening parabola is below the vertex of the downward-opening parabola, the graphs intersect at two points.This can also be true when a and n have the same sign. y y

–4 –2 O 2 4 x –4 –2 O 2 4 x –2 –2

–4 –4

Infinitely many solutions: If the equations are equal there are infinitely many points of intersection. 197. If a linear factor of a polynomial is repeated, then the zero is repeated and the function has multiple zeros. To determine whether a function has a multiple zero, factor the polynomial. If a factor is repeated in the factored expression, then it is a multiple zero. 198. ÷ (x – 3) = with no remainder, so x = 3 is a real zero of the function. 199. By the Irrational Root Theorem, if is a root, then its conjugate – is also a root. If is a root, then its conjugate is also a root. Two additional roots are – and . 200. First, multiply the numerator and denominator by so the denominator becomes a whole number.

Apply the Distributive Property to the numerator to simplify.

Now, divide each term of the numerator by the denominator of 5 to get . 201. a. Let x = the original price Cost with 25% discount: f(x) = x – 0.25x = 0.75x Cost with a coupon for $30: g(x) = x – 30

b. c. d. It will cost $7.50 more. e. If the coupon is applied first, then the cost of the item is less. Therefore, the discount is smaller than it would be on the higher original price.