Deck 3: Quadratic Functions and Equations; Inequalities

Question 1

Match the equation to the correct graph.
$$y = 2 ( x + 2 ) ^ { 2 } - 4$$

A)

B)

C)

D)

Answer 1

B

Question 2

Solve by completing the square to obtain exact solutions.
$$x ^ { 2 } + 4 x + 53 = 0$$

A) $- 2 \pm \sqrt { 53 } \mathrm { i }$
B) $- 2 \pm 7 \mathrm { i }$
C) $5 , - 9$
D) $2 \pm 7 \mathrm { i }$

Answer 2

B

Question 3

Match the equation to the correct graph.
$$y = - ( x + 6 ) ^ { 2 } + 3$$

A)

B)

C)

D)

Answer 3

D

Question 4

Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
$$f ( x ) = x ^ { 2 } - 5 x + 6$$

A) Increasing on $\left( \frac { 5 } { 2 } , \infty \right)$ ; decreasing on $\left( - \infty , \frac { 5 } { 2 } \right)$
B) Increasing on $\left( - \frac { 5 } { 2 } , \infty \right) ;$ decreasing on $\left( - \infty , - \frac { 5 } { 2 } \right)$
C) Increasing on $\left( - \infty , - \frac { 5 } { 2 } \right) ;$ decreasing on $\left( - \frac { 5 } { 2 } , \infty \right)$
D) Increasing on $\left( - \infty , \frac { 5 } { 2 } \right)$ ; decreasing on $\left( \frac { 5 } { 2 } , \infty \right)$

Question 5

Find the vertex of the parabola.
$$f ( x ) = 2 x ^ { 2 } - 12 x + 16$$

A) $( - 3,2 )$
В) $( 2 , - 3 )$
C) $( 3 , - 2 )$
D) $( - 2,3 )$

Question 6

Match the equation to the correct graph.
$$y = - \frac { 1 } { 3 } ( x + 5 ) ^ { 2 } - 5$$

A)

B)

C)

D)

Question 7

Match the equation to the correct graph.
$$y = - ( x + 3 ) ^ { 2 } + 2$$

A)

B)

C)

D)

Question 8

Match the equation to the correct graph.
$$y = 2 ( x - 2 ) ^ { 2 } - 3$$

A)

B)

C)

D)

Question 9

State whether the function is linear or quadratic.
$$f ( x ) = - 3 x ^ { 2 }$$

A) Linear
B) Quadratic

Question 10

Solve.
$$6 x ^ { 3 } + x ^ { 2 } - 6 x - 1 = 0$$

A) $1 , - 1 , - \frac { 1 } { 6 }$
B) $1 , - 1$
C) $- 1,6$
D) $1 , \frac { 1 } { 6 }$

Question 11

Match the equation to the correct graph.
$$y = x ^ { 2 } - 7$$

A)

B)

C)

D)

Question 12

Match the equation to the correct graph.
$$y = ( x + 6 ) ^ { 2 }$$

A)

B)

C)

D)

Question 13

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$( 9 - 7 \mathrm { i } ) ( 8 - 5 \mathrm { i } )$

A) $35 \mathrm { i } ^ { 2 } - 101 \mathrm { i } + 72$
B) $107 - 11 \mathrm { i }$
C) $37 + 101 \mathrm { i }$
D) $37 - 101 \mathrm { i }$

Question 14

Solve the inequality and graph the solution set.
$$| 5 x + 5 | < - 2$$

A)

B) No solution
11ecbb04_e349_f208_8bc5_b79e5e3efd35_TB2705_11
C)

D)

Question 15

Solve.
$$\sqrt [ 5 ] { 6 x + 7 } = 3$$

A) 243
B) $\frac { 118 } { 3 }$
C) $\frac { 81 } { 2 }$
D) 236

Question 16

Match the equation to the correct graph.
$$y = ( x - 3 ) ^ { 2 }$$

A)

B)

C)

D)

Question 17

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$$4 \mathrm { i } ( 9 - 4 \mathrm { i } )$$

A) $36 \mathrm { i } - 16 \mathrm { i } ^ { 2 }$
В) $- 16 + 36 i$
C) $16 + 36 \mathrm { i }$
D) $16 - 36 \mathrm { i }$

Question 18

Match the equation to the correct graph.
$$y = \frac { 1 } { 2 } ( x - 4 ) ^ { 2 } + 5$$

A)

B)

C)

D)

Question 19

Solve.
$\sqrt { 10 q - 9 } = 9$

A) 81
B) $\frac { 81 } { 10 }$
C) 9
D) $\frac { 36 } { 5 }$

Question 20

Consider only the discriminant, $\mathrm { b } ^ { 2 }$ 4ac, to determine whether one real-number solution, two different real-
solutions, or two different imaginary-number solutions exist.
$$x ^ { 2 } + 10 x + 25 = 0$$

A) Two different real-number solutions
B) Two different imaginary-number solutions
C) One real solution

Question 21

Find the axis of symmetry of the given function.
$$f ( x ) = - 2 x ^ { 2 } + 13 x + 4$$

A) $x = - \frac { 13 } { 2 }$
B) $x = - \frac { 13 } { 4 }$
C) $x = \frac { 13 } { 4 }$
D) $x = \frac { 13 } { 2 }$

Question 22

Express the number in terms of i.
$\sqrt { - 9 }$

A) $\sqrt { 3 } \mathrm { i }$
B) $- \sqrt { 3 } \mathrm { i }$
C) $- 3 \mathrm { i }$
D) $3 \mathrm { i }$

Question 23

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$$( 3 + 3 i ) ( 3 - 3 i )$$

A) $9 - 9 \mathrm { i }$
B) $9 + 9 \mathrm { i }$
C) 0
D) 18

Question 24

State whether the function is linear or quadratic.
$$f ( x ) = - 9 x - 1$$

A) Linear
B) Quadratic

Question 25

Solve.
$$x ^ { 2 / 3 } - 7 x ^ { 1 / 3 } + 10 = 0$$

A) $- 8,25$
B) $\pm 1,125$
C) 8,125
D) $\pm 8 , \pm 125$

Question 26

Solve.
$$| 8 x | = 2$$

A) $- \frac { 1 } { 4 } , \frac { 1 } { 4 }$
B) $- 4,4$
C) $- \frac { 1 } { 4 }$
D) $\frac { 1 } { 4 }$

Question 27

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$$( 8 + 2 \mathrm { i } ) ^ { 2 }$$

A) $68 - 32 \mathrm { i }$
B) $60 - 32 \mathrm { i }$
C) $68 + 32 \mathrm { i }$
D) $60 + 32 \mathrm { i }$

Question 28

Solve.
The length and width of a rectangle have a sum of 84. What dimensions give the maximum area?

A)Length 32 and width 52
B)Length 41 and width 43
C)Length 42 and width 42
D)Length 33 and width 51

Question 29

Simplify. Write your answer in the form a + bi, where a and b are real numbers.
$$( 19 + 6 i ) - ( 7 + 3 i )$$

A) $- 12 - 3 i$
B) $12 + 3 i$
C) $- 12 + 3 i$
D) $12 - 3 \mathrm { i }$

Question 30

Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
$$f ( x ) = 2 x ^ { 2 } + 8 x + 6$$

A) Increasing on $( - 4 , \infty )$ ; decreasing on $( - \infty , - 4 )$
B) Increasing on $( 8 , \infty )$ ; decreasing on $( - \infty , 8 )$
C) Increasing on $( 2 , \infty )$ ; decreasing on $( - \infty , 2 )$
D) Increasing on $( - 2 , \infty )$ ; decreasing on $( - \infty , - 2 )$

Question 31

State whether the function is linear or quadratic.
$$f ( x ) = 2 x$$

A) Linear
B) Quadratic

Question 32

Use a graphing calculator to find the zeros of the function. Round to three decimal places.
$$f ( x ) = 5 x ^ { 2 } + 2 x - 1$$

A) $- 0.424,0.824$
B) $- 0.824,0.424$
C) $- 0.290,0.690$
D) $- 0.690,0.290$

Question 33

Use the given graph to find the x-intercepts and zeros of the function.

A) $( - 7,0 ) , ( 4,0 ) ; - 7,4$
B) $( 4,0 ) , ( 7,0 ) ; 4,7$
C) $( - 4,0 ) , ( 7,0 ) ; - 4,7$
D) $( - 4,0 ) , ( - 7,0 ) ; - 4 , - 7$

Question 34

Solve graphically. Round solutions to three decimal places, where appropriate.
$$5 x ^ { 2 } - 20 x + 11 = 0$$

A) $1.158,3.842$
B) $- 5.175 , - 2.492$
C) $- 3.675 , - 0.992$
D) $0.658,3.342$

Question 35

Solve the inequality and graph the solution set.
$$| x | > 5$$

A)

B)

C)

D)

Question 36

Solve.
$$\frac { 1 } { x - 6 } - \frac { 1 } { x } = \frac { 6 } { x ^ { 2 } - 6 x }$$

A) $\pm \sqrt { 6 } \mathrm { i }$
B) $\pm 6 \mathrm { i }$
C) $\{ x \mid x$ is a real number and $x \neq 0$ and $x \neq 6 \}$
D) No solution

Question 37

Solve the inequality and graph the solution set.
$$\left| \frac { 3 x + 6 } { 2 } \right| > 3$$

A)

B)

C)

D)

Question 38

Solve.
$$( 4 x - 7 ) ( 4 x - 3 ) = 0$$

A) $- \frac { 7 } { 4 } , - \frac { 3 } { 4 }$
B) $\frac { 7 } { 4 } , \frac { 3 } { 4 }$
C) $\frac { 7 } { 16 } , \frac { 7 } { 16 }$
D) $\frac { 3 } { 8 } , \frac { 9 } { 8 }$

Question 39

Solve the problem.
The area of a square is 81 square centimeters. If the same amount is added to one dimension and removed from the other, the resulting rectangle has an area 9 square centimeters less than the area
Of the square. How much is added and subtracted?

A)12 cm
B)3 cm
C)4 cm
D)9 cm

Question 40

Express the number in terms of i.
$- \sqrt { - 75 }$

A) $- 3 \sqrt { 5 } \mathrm { i }$
B) $- 15 \mathrm { i }$
C) $- 5 \sqrt { 3 } \mathrm { i }$
D) $5 \sqrt { 3 } \mathrm { i }$

Question 41

Solve by completing the square to obtain exact solutions.
$$9 x ^ { 2 } + 27 x + 20 = 0$$

A) $- \frac { 3 } { 4 } , - \frac { 5 } { 3 }$
B) $- \frac { 4 } { 3 } , - \frac { 5 } { 3 }$
C) $\frac { 4 } { 3 } , \frac { 5 } { 3 }$
D) $\frac { 3 } { 4 } , \frac { 3 } { 5 }$

Question 42

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$$\frac { 1 + 2 i } { ( 1 - 2 i ) ^ { 2 } }$$

A) $- \frac { 11 } { 25 } - \frac { 1 } { 5 } \mathrm { i }$
B) $\frac { 1 } { 25 } + \frac { 6 } { 25 }$ i
C) $- \frac { 11 } { 25 } + \frac { 14 } { 25 } \mathrm { i }$
D) $- \frac { 11 } { 25 } - \frac { 2 } { 25 } \mathrm { i }$

Question 43

Solve.
$$\left| \frac { 1 } { 5 } x - 7 \right| = 4$$

A) 55
B) 15,55
C) 15
D) $15,55,0$

Question 44

Solve.
$$m ^ { 2 / 5 } + 2 m ^ { 1 / 5 } - 24 = 0$$

A) 1024
B) $1024 , - 7776$
C) $- 1024,7776$
D) $1,048,576,60,466,176$

Question 45

Solve and write interval notation for the solution set.
$$| 3 x - 7 | > - 6$$

A) No solution
B) $( - \infty , \infty )$
C) $\left( - \infty , \frac { 1 } { 3 } \right) \cup \left( - \frac { 11 } { 3 } , \infty \right)$
D) $\left( - \infty , \frac { 13 } { 3 } \right) \cup \left( \frac { 1 } { 3 } , \infty \right)$

Question 46

Solve the inequality and graph the solution set.
$$| x | \geq 7$$

A)

B)

C)

D)

Question 47

State whether the function is linear or quadratic.
$$f ( x ) = 0.6 x ^ { 2 } + 10 x + 21$$

A) Linear
B) Quadratic

Question 48

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$$( 9 - 5 i ) ( 2 + 4 i )$$

A) $38 - 26 \mathrm { i }$
B) $38 + 26 i$
C) $- 2 - 46 \mathrm { i }$
D) $- 20 i ^ { 2 } + 26 i - 18$

Question 49

Solve the problem.
A stuntman jumps from a rooftop $370 \mathrm { ft }$ off the ground. How long will it take him, falling freely, to reach the ground? (Use the formula s $= 16 \mathrm { t } ^ { 2 }$ .)

A) About $11.6 \mathrm { sec }$
B) About $23.1 \mathrm { sec }$
C) About $4.8 \mathrm { sec }$
D) About $76.9 \mathrm { sec }$

Question 50

Find the zeros of the function. Give exact answers.
$$f ( x ) = 3 x ^ { 2 } - x + 6$$

A) $\frac { 1 } { 3 } \pm \frac { \sqrt { 71 } } { 3 }$ i
B) $1 \pm \sqrt { 71 }$
C) $\frac { 1 } { 6 } \pm \frac { \sqrt { 71 } } { 6 } \mathrm { i }$
D) $- \frac { 1 } { 3 } , \frac { 1 } { 6 }$

Question 51

Find the range of the given function.
$$f ( x ) = 3 x ^ { 2 } + 24 x + 51$$

A) $[ 3 , \infty )$
B) $( - \infty , 4 ]$
C) $( - \infty , - 3 ]$
D) $[ - 4 , \infty )$

Question 52

Simplify.
$\mathrm { i } ^ { 32 }$

A) 1
B) -1
C) $- \mathrm { i }$
D) $\mathrm { i }$

Question 53

Use the given graph to find the x-intercepts and zeros of the function.

A) $( - 3,0 ) , ( 5,0 ) ; - 3,5$
B) $( 3,0 ) , ( 5,0 ) ; 3,5$
C) $( - 5,0 ) , ( - 3,0 ) ; - 5 , - 3$
D) $( - 5,0 ) , ( 3,0 ) ; - 5,3$

Question 54

Solve.
$$| x | = 3.2$$

A) $- 3.2,3.2$
B) 1024
C) $- 3.2$
D) $3.2$

Question 55

Solve. $\sqrt { 2 x + 3 } - \sqrt { x + 1 } = 1$
A) $- 3 , - 1$
B) $3 , - 1$
C) 3
D) No solution
Solve. $\sqrt { 2 x + 3 } - \sqrt { x + 1 } = 1$

A) $- 3 , - 1$
B) $3 , - 1$
C) 3
D) No solution

Question 56

Find the zeros of the function. Give exact answers.
$$f ( x ) = x ^ { 2 } - 5 x + 1$$

A) $\frac { 5 \pm \sqrt { 21 } } { 2 }$
B) $\frac { 5 \pm \sqrt { 29 } } { 2 }$
C) $\frac { - 5 \pm \sqrt { 21 } \mathrm { i } } { 2 }$
D) $\frac { 5 \pm \sqrt { 29 } \mathrm { i } } { 2 }$

Question 57

Find the axis of symmetry of the given function.
$$f ( x ) = 3 x ^ { 2 } + 24 x + 53$$

A) $x = - 4$
B) $x = - 5$
C) $x = 4$
D) $x = 5$

Question 58

Find the zeros of the function. Give exact answers.
$$f ( x ) = x ^ { 2 } - 5 x - 8$$

A) $\frac { 5 \pm \sqrt { 57 } } { 2 }$
B) $\frac { - 5 \pm \sqrt { 57 } i } { 2 }$
C) $\frac { - 25 \pm \sqrt { 57 } } { 2 }$
D) $\frac { 15 \pm \sqrt { 57 } } { 2 }$

Question 59

Find the vertex of the parabola.
$$f ( x ) = - 3 x ^ { 2 } + 24 x - 51$$

A) $( 4 , - 3 )$
В) $( - 4,3 )$
C) $( - 3,4 )$
D) $( 3 , - 4 )$

Question 60

Solve.
$$56 x + 8 x ^ { 2 } = 0$$

A) $- 7$
B) $0 , - 7$
C) $- 8 , - 7$
D) $- 28$

Question 61

Solve.
$$| x + 2 | = 4$$

A) No solution
B) $- 2.6$
D) $2 , - 6$

Question 62

Solve and write interval notation for the solution set.
$$| 4 x | < 3$$

A) $\left( - \infty , \frac { 3 } { 4 } \right)$
B) $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$
C) $\left( - \frac { 4 } { 3 } , \frac { 4 } { 3 } \right)$
D) $\left( - \infty , \frac { 4 } { 3 } \right)$

Question 63

Solve.
John owns a hotdog stand. He has found that his profit is represented by the function $\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 74 \mathrm { x } + 80$ , with $\mathrm { P } ( \mathrm { x } )$ being profits and $x$ the number of hotdogs sold. How many hotdogs must he sell to earn the most profit?

A) 21 hotdogs
B) 43 hotdogs
C) 38 hotdogs
D) 37 hotdogs

Question 64

Solve.
$$\sqrt { 3 x + 14 } = x + 3$$

A) 0
B) $\frac { - 3 \pm \sqrt { 29 } } { 2 }$
C) $\frac { - 3 + \sqrt { 29 } } { 2 }$
D) $\frac { - 3 - \sqrt { 29 } } { 2 }$

Question 65

Solve.
$$x ^ { 2 } + 2 x - 48 = 0$$

A) $- 40 , - 8$
B) $\sqrt { - 48 } , - \sqrt { - 48 }$
C) $- 6,8$
D) $6 , - 8$

Question 66

Determine whether there is a maximum or minimum value for the given function, and find that value.
$$f ( x ) = - x ^ { 2 } - 8 x - 26$$

A) Minimum: 0
B) Minimum: 10
C) Maximum: 10
D) Maximum: $- 10$

Question 67

Consider only the discriminant, $b ^ { 2 } - 4 a c$ , to determine whether one real-number solution, two different real-number solutions, or two different imaginary-number solutions exist.
$$x ^ { 2 } - 2 x - 6 = 0$$

A)Two different imaginary-number solutions
B)One real solution
C)Two different real-number solutions

Question 68

Find the range of the given function.
$$f ( x ) = x ^ { 2 } - 14 x + 58$$

A) $[ 7 , \infty )$
B) $( - \infty , 0 ]$
C) $[ - 7 , \infty )$
D) $[ 9 , \infty )$

Question 69

Find the zeros of the function. Give exact answers.
$$f ( x ) = x ^ { 2 } - 9 x + 3$$

A) $\frac { - 9 \pm \sqrt { 69 } \mathrm { i } } { 2 }$
B) $\frac { 9 \pm \sqrt { 69 } } { 2 }$
C) $\frac { 27 \pm \sqrt { 69 } } { 2 }$
D) $\frac { - 9 \pm \sqrt { 69 } } { 4 }$

Question 70

Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
$$f ( x ) = x ^ { 2 } - 8 x + 7$$

A) Increasing on $( 4 , \infty )$ ; decreasing on $( - \infty , 4 )$
B) Increasing on $( - 4 , \infty )$ ; decreasing on $( - \infty , - 4 )$
C) Increasing on $( - \infty , - 4 )$ ; decreasing on $( - 4 , \infty )$
D) Increasing on $( - \infty , 4 )$ ; decreasing on $( 4 , \infty )$

Question 71

Solve.
$12 x ^ { 3 } - 30 x ^ { 2 } + 10 x - 25 = 0$

A) $\frac { 2 } { 5 }$
B) 5
C) $\frac { 5 } { 2 }$
D) $- \frac { 5 } { 2 }$

Question 72

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$- 6 i ( - 2 + 5 i )$

A) $30 - 12 \mathrm { i }$
B) $12 \mathrm { i } - 30 \mathrm { i } ^ { 2 }$
C) $- 30 - 12 \mathrm { i }$
D) $30 + 12 \mathrm { i }$

Question 73

Solve graphically. Round solutions to three decimal places, where appropriate.
$$6 x ^ { 2 } - 15 x + 7 = 0$$

A) $- 0.379,0.879$
B) $0.721,1.979$
C) $1.121,2.379$
D) $0.621,1.879$

Question 74

Solve.
$$\frac { 1 } { t } + \frac { 1 } { 3 t } + \frac { 1 } { 5 t } = 9$$

A) $\frac { 23 } { 135 }$
B) $\frac { 23 } { 45 }$
C) $\frac { 19 } { 135 }$
D) $\frac { 23 } { 27 }$

Question 75

Solve.
$$\sqrt [ 3 ] { x + 3 } = 4$$

A) 64
B) 1
C) 13
D) 61

Question 76

Simplify.
$( 3 i ) ^ { 5 }$

A) $- 243$
B) $243 \mathrm { i }$
C) 243
D) $- 243 \mathrm { i }$

Question 77

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$( 6 - \sqrt { - 36 } ) ( 2 + \sqrt { - 100 } )$

A) $72 + 48 \mathrm { i }$
B) $72 - 48 \mathrm { i }$
C) $- 72 + 48 i$
D) $3612 + 672 \mathrm { i }$

Question 78

Determine whether there is a maximum or minimum value for the given function, and find that value.
$$f ( x ) = x ^ { 2 } + 6 x - 1$$

A) Maximum: $- 10$
B) Maximum: 0
C) Minimum: $- 3$
D) Minimum: $- 10$

Question 79

Simplify.

- $i^{10}$

A) $- \mathrm { i }$
B) 1
C) $\mathrm { i }$
D) $- 1$

Question 80

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
$$( 6 + 6 i ) ( 6 + 4 i )$$

A) $60 + 12 \mathrm { i }$
B) $12 + 60 \mathrm { i }$
C) $12 - 60 \mathrm { i }$
D) $24 \mathrm { i } ^ { 2 } + 60 \mathrm { i } + 36$