Name: ________________________ Class: ___________________ Date: __________ ID: A

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Unit 3 quadratic functions review

Multiple ChoiceIdentify the choice that best completes the statement or answers the question.

____ 1. What is the vertex of y = 7(x + 5)2 + 4?

A (5, 4) C (–5, 4)B (–4, 5) D (7, –4)

____ 2. Which graph represents the quadratic function y = 57

(x − 4)2 − 7?

A C

B D

____ 3. The vertex of a parabola is located at (−5, 6). If the parabola has a y-intercept of 231, which quadratic function represents the parabola?

A f x( ) = 9(x − 5)2 + 6 C f x( ) = −9(x + 5)2 + 6

B f x( ) = 9(x + 5)2 + 6 D f x( ) = 9(x − 5)2 − 6

Name: ________________________ ID: A

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____ 4. Which function is not quadratic?

A f x( ) = (6x + 9)19

x − 9Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ C f x( ) = 7x2 + 8

B f x( ) = x(x − 9)(6x + 8) D f x( ) = 6(x − 9)2

____ 5. Identify the characteristics of this graph.

A vertex: (–2, –5)axis of symmetry: x = −2y-intercept: 10.5x-intercepts: –3 and –7opens downward

C vertex: (–2, –5)axis of symmetry: x = −2y-intercept: 10.5x-intercepts: –3 and –7opens upward

B vertex: (–5, –2)axis of symmetry: x = −5y-intercept: 10.5x-intercepts: –3 and –7opens upward

D vertex: (–5, –2)axis of symmetry: x = −2y-intercept: 10.5x-intercepts: 3 and 7opens downward

Name: ________________________ ID: A

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____ 6. What is g x( ) = (−2x + 6)(4x − 14) written in standard form?

A g(x) = −8x2 + 28x + 8 C g(x) = 2x2 − 8

B g(x) = −8x2 + 52x − 84 D g(x) = −8x2 − 84

____ 7. What are the coordinates of the vertex of the quadratic function y = 4x2 + 8x − 2?

A (–6, –1) C (–1, –6)B (8, –2) D (8, –6)

____ 8. What is the function y = 2(x − 4)2 − 2 written in standard form?

A y = 2x2 − 8x + 30 C y = 2x2 − 16x + 34

B y = 2x2 − 8x + 34 D y = 2x2 − 16x + 30

____ 9. What is the equation of the quadratic function y = x2 + 24x + 29 in vertex form?

A y = (x + 12)2 − 173 C y = (x − 12)2 − 173

B y = (x − 12)2 − 115 D y = (x + 12)2 − 115

____ 10. What is the equation of the quadratic function y = x2 − 26x + 41 in vertex form?

A y = −(x + 13)2 − 210 C y = (x + 13)2 − 128

B y = −(x − 13)2 − 210 D y = (x − 13)2 − 128

____ 11. Which quadratic function in standard form represents y = 3(x − 1)2 − 25?

A y = 3x2 − 3x − 11 C y = 3x2 + 6x − 22

B y = 3x2 − 6x − 22 D y = 3x2 − 6x − 11

____ 12. State whether the function y = 4x2 − 36x − 43 has a maximum or minimum value and identify the coordinates of the vertex.

A maximum at (4.5,− 124) C minimum at (−124, 4.5)B maximum at (−124, 4.5) D minimum at (4.5,− 124)

____ 13. The vertex of the quadratic function y = −7 / 9( )x2 − 1 / 6( )x − 1 / 81 is

A −3 / 28,−31 / 9072ÊËÁÁ ˆ

¯˜̃ C 31 / 9072,−3 / 28Ê

ËÁÁ ˆ

¯˜̃

B −31 / 9072,−3 / 28ÊËÁÁ ˆ

¯˜̃ D 3 / 28,−31 / 9072Ê

ËÁÁ ˆ

¯˜̃

Name: ________________________ ID: A

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CompletionComplete each statement.

1. A quadratic function with vertex (0, 1) and two x-intercepts will open _______________.

2. The quadratic function in vertex form that represents the graph shown below is ____________________ .

Name: ________________________ ID: A

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Short Answer

1. Sketch the graph of the quadratic relation y = −x2 − 9x + 10. Label the x-intercepts and the vertex.

2. Express the quadratic function y = −3x2 + 12x − 10 in vertex form.

Name: ________________________ ID: A

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Problem

1. a) Write the function y = –(x – 2)2 + 9 in standard form.b) Sketch the graph of the function. Use your answer to part a) to identify the y-intercept.

2. Consider the function y = –2(x – 12)2 + 18.a) What is the axis of symmetry of the graph of the function.b) What is the vertex of the graph of the function?c) State the domain and range of the function.

3. A store can increase its profit by increasing the price of the sweaters it sells. The relation between the income, R, and the dollar increase in the price per sweater, d, can be modelled by the equation

R = −50 d − 3.5( )2 + 4500.

a) What is the maximum possible income?b) What would the income be if the price per sweater were increased by $10?

4. A baseball batter hits an infield fly ball. The height, h, in metres, of the baseball after t seconds is approximately modelled by the function h(t) = –5t2 + 4t + 1.a) State the domain and range of the function.b) What is the initial height of the ball?c) How long does it take for the ball to hit the ground?

Name: ________________________ ID: A

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5. On a forward somersault dive, Nina’s height, h, in metres, above the water is approximately modelled by the

function h = −5t2 + 7t + 4, where t is the time, in seconds, after she leaves the diving board. Graph the function and use the graph to complete the following.a) Find Nina’s maximum height above the water.b) How long does it take her to reach the maximum height?c) How long is it before she enters the water?d) How high is the board above the water?

ID: A

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Unit 3 quadratic functions reviewAnswer Section

MULTIPLE CHOICE

1. ANS: C PTS: 1 DIF: Average OBJ: Section 3.1NAT: RF 3 TOP: Investigating Quadratic Functions in Vertex FormKEY: vertex

2. ANS: B PTS: 1 DIF: Difficult OBJ: Section 3.1NAT: RF 3 TOP: Investigating Quadratic Functions in Vertex FormKEY: vertex form | graph

3. ANS: B PTS: 1 DIF: Difficult OBJ: Section 3.1NAT: RF 3 TOP: Investigating Quadratic Functions in Vertex FormKEY: vertex | y-intercept

4. ANS: B PTS: 1 DIF: Easy OBJ: Section 3.2NAT: RF 4 TOP: Investigating Quadratic Functions in Standard FormKEY: identify quadratic functions

5. ANS: B PTS: 1 DIF: Average OBJ: Section 3.2NAT: RF 4 TOP: Investigating Quadratic Functions in Standard FormKEY: vertex | axis of symmetry | y-intercept | x-intercept | direction of opening

6. ANS: B PTS: 1 DIF: Easy OBJ: Section 3.2NAT: RF 4 TOP: Investigating Quadratic Functions in Standard FormKEY: standard form

7. ANS: C PTS: 1 DIF: Average OBJ: Section 3.2NAT: RF 4 TOP: Investigating Quadratic Functions in Standard FormKEY: vertex

8. ANS: D PTS: 1 DIF: Average OBJ: Section 3.2NAT: RF 4 TOP: Investigating Quadratic Functions in Standard FormKEY: standard form

9. ANS: D PTS: 1 DIF: Easy OBJ: Section 3.3NAT: RF 4 TOP: Completing the Square KEY: standard to vertex form

10. ANS: D PTS: 1 DIF: Average OBJ: Section 3.3NAT: RF 4 TOP: Completing the Square KEY: standard to vertex form

11. ANS: B PTS: 1 DIF: Average OBJ: Section 3.3NAT: RF 4 TOP: Completing the Square KEY: vertex to standard form

12. ANS: D PTS: 1 DIF: Difficult OBJ: Section 3.3NAT: RF 4 TOP: Completing the Square KEY: max/min

13. ANS: A PTS: 1 DIF: Difficult OBJ: Section 3.3NAT: RF 4 TOP: Completing the Square KEY: vertex | fraction

COMPLETION

1. ANS: downward

PTS: 1 DIF: Easy OBJ: Section 3.2 NAT: RF 4TOP: Investigating Quadratic Functions in Standard Form KEY: direction of opening

ID: A

2

2. ANS: y = −4(x + 3)2 − 5

PTS: 1 DIF: Average OBJ: Section 3.1 NAT: RF 3TOP: Investigating Quadratic Functions in Vertex Form KEY: vertex form

SHORT ANSWER

1. ANS:

PTS: 1 DIF: Average OBJ: Section 3.2 NAT: RF 4TOP: Investigating Quadratic Functions in Standard Form KEY: standard form | graph | vertex | x-intercepts

ID: A

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2. ANS:

y = −3x2 + 12x − 10

= −3 x2 − 4xÊËÁÁÁ

ˆ¯˜̃̃ − 10

= −3(x2 − 4x + 4− 4) − 10

= −3((x2 − 4x + 4) − 4) − 10

= −3((x − 2)2 − 4) − 10

= −3(x − 2)2 + 12− 10

= −3(x − 2)2 + 2

PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF 4TOP: Completing the Square KEY: standard to vertex form

ID: A

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PROBLEM

1. ANS:

a) y = −(x − 2)2 + 9

= −(x2 − 4x + 4) + 9

= −x2 + 4x − 4+ 9

= −x2 + 4x + 5b) The y-intercept is 5.

PTS: 1 DIF: Average OBJ: Section 3.1 NAT: RF 3TOP: Investigating Quadratic Functions in Vertex Form KEY: vertex to standard form | vertex

2. ANS: a) x = 12b) (12, 18)c) Domain: (−∞, ∞) Range: (−∞, 18]

PTS: 1 DIF: Easy OBJ: Section 3.1 NAT: RF 3TOP: Investigating Quadratic Functions in Vertex Form KEY: axis of symmetry | domain | range | vertex

ID: A

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3. ANS: a) The maximum profit occurs at the vertex (3.5, 4500) or $4500.b) Substitute d = 10 into the equation:

R = −50 10− 3.5( )2 + 4500

= −50 6.5( )2 + 4500

= −2112.50+ 4500

= 2387.5The income would be $2387.50.

PTS: 1 DIF: Average OBJ: Section 3.1 NAT: RF 3TOP: Investigating Quadratic Functions in Vertex Form KEY: vertex form | vertex

4. ANS: a) Find the t-intercepts to determine the domain.

h = −5t2 + 4t + 1

= (−5t − 1)(t − 1)

The t-intercepts are −15

and 1. Since t ≥ 0, the domain is { t ∈ R | 0 ≤ t ≤ 1} .

To find the range, write the equation in vertex form.

h = −5t2 + 4t + 1

= −5(t2 − 0.8t) + 1

= −5(t2 − 0.8t + 0.16− 0.16)+ 1

= −5(t − 0.4)2 + 0.8+ 1

= −5(t − 0.4)2 + 1.8The parabola opens downward, so the maximum value is the h-coordinate of the vertex, or 1.8. Thus, the range is { h(t) ∈ R| 0 ≤ h(t) ≤ 1.8}b) The initial height of the ball is the h-intercept, or 1 m.c) The time it takes for the ball to hit the ground is the t-intercept that is greater than zero, or 1 s.

PTS: 1 DIF: Easy OBJ: Section 3.2 NAT: RF 4TOP: Investigating Quadratic Functions in Standard Form KEY: domain | range | intercept

ID: A

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5. ANS: Graph the function for t ≥ 0 using technology.

a) From the graph, the maximum height is about 6.5 m.b) It takes about 0.7 s to reach the maximum height.c) The t-intercept is approximately 1.8. It takes Nina about 1.8 s to enter the water.d) Substituting t = 0 into the equation, or reading from the graph at t = 0, h = 4. So, the board is 4 m above the water.

PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF 4TOP: Completing the Square KEY: standard form | modelling

ID: A Unit 3 quadratic functions review [Answer Strip]

_____ 1.C

_____ 2.B

_____ 3.B

_____ 4.B

_____ 5.B

_____ 6.B

_____ 7.C

_____ 8.D

_____ 9.D

_____10.D

_____11.B

_____12.D

_____13.A