Chapter 5 Diagnostic Test - Nelson

Chapter 5 Diagnostic Test STUDENT BOOK PAGES 244–307

1. a) Complete this table of values for the relation y = 2x2 – 5. x –3 –2 –1 0 1 3

y 13 3 b) Plot the points, and sketch the graph.

2. a) Sketch the graph of the relation y = 2x – 3. b) Sketch the reflection of y = 2x – 3 in the x-axis. c) Determine the equation of the line you sketched for part b).

3. a) Determine the zeros of the relation y = 3(x – 1)(x – 5). b) Is the relation quadratic? Explain how you know. c) Sketch the relation. d) Determine the coordinates of the vertex of the relation, if there is a vertex, or explain

why this is impossible to do.

4. For each translation, determine a) the vertices of +ABC after a translation 3 units left and 2 units up b) the equation of the relation y = –x + 6 after a translation 2 units up c) the coordinates of the points (–2, 4), (–1, 1), (0, 0), and (1, 1) after a translation

that moves (2, 4) to (4, 7)

5. Ricardo performed the following calculation: 3(4 – 2)2 +7 = 3(2)2 + 7 = 62 + 7 = 36 + 7 = 43 What error did Ricardo make?

6. Braking distances for a pickup truck that is travelling at different speeds are shown in this table. Speed (km/h) 30 45 60 75 90 105 120

Braking Distance (m) 22 45 74 105 145 211 252

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a) Use the data to create a scatter plot. b) Sketch a line of good fit for the data. How well does the line appear to fit the data? c) Sketch a curve of good fit for the data. How well does the curve appear to fit the data?

What kind of relation might the curve be?

7. Factor, if possible. a) x2 + 2x – 3 b) 4x2 – 9 c) 2x2 + 7x + 15 d) x2 – 6x + 8

Chapter 5 Diagnostic Test | 1

Chapter 5 Diagnostic Test Answers

1. a) x –3 –2 –1 0 1 2 3

y 13 3 –3 –5 –3 3 13

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b)

2. a), b)

c) y = –2x + 3

3. a) x = 1, x = 5 b) yes, because expanding gives

y = 3x2 – 18x + 15

c) d) (3, –12)

4. a) A → (–2, 8), B → (–1, 4), C → (2, 6) b) y = –x + 8 c) (0, 7), (1, 4), (2, 3), (3, 4)

5. Ricardo should have evaluated the exponent (2)2 and then multiplied his result by 3, rather than multiplying 2 by 3.

6. a)–c)

b) The line fits the data reasonably well. c) The curve fits the data reasonably well. The

curve might be a quadratic relation.

7. a) (x – 1)(x + 3) c) cannot be factored b) (2x – 3)(2x + 3) d) (x – 2)(x – 4)

If students have difficulty with the questions in the Diagnostic Test, you may need to review the following topics: • creating tables of values and using them to draw graphs • performing translations on the Cartesian coordinate plane • applying the order of operations • creating a scatter plot and drawing a line of good fit • factoring a quadratic relation

2 | Chapter 5 Diagnostic Test Answers

Lesson 5.1 Extra Practice STUDENT BOOK PAGES 250–258

1. Match each graph with the correct equation. a) y = 0.5x2 c) y = –2x2

b) y = –x2 d) y = 31 x2

i) iii)

ii) iv)

2. Write the equation of each graph. a) b)

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d.

3. Determine the equation of a quadratic model to represent the lower arch of this bridge.

4. Which of the following relations do not represent parabolas that are vertical compressions of the graph of y = x2? a) y = 0.1x2 c) y = –2x2

b) y = x2 d) y = 51

− x2

5. Sketch the parabolas for question 4 that are vertical compressions of the graph of y = x2.

6. What transformation(s) must be applied to the graph of y = x2 to create each parabola shown? a) b)

7. The graph of y = x2 is vertically compressed by

a factor of 32 and reflected in the x-axis. What

is the equation of the resulting graph?

Lesson 5.1 Extra Practice | 3

Lesson 5.1 Extra Practice Answers 1. a) ii) c) i) b) iv) d) iii)

2. a) y = 3x2 b) y = 41

− x2

3. y = –0.125x2

4. b) and c)

5. a)

d)

6. a) reflection in the x-axis, vertical stretch by a factor of 5

b) vertical compression by a factor of 0.75

7. y = 2

32 x−

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4 | Principles of Mathematics 10: Lesson 5.1 Extra Practice Answers


1. List, in order, the transformations you would apply to the graph of y = x2 to create the graph of each relation.

a) y = 2x2 + 1 c) y = 2

21 x – 4

b) y = (x – 3)2 d) y = 3(x + 3)2 – 6

2. Sketch the graph of each relation in question 1.

3. Identify the vertex of each parabola. a) y = (x – 3)2 + 2 c) y = 2x2 – 5

b) y = (x + 7)2 d) y = 43

41

31 2

−⎟⎠⎞

⎜⎝⎛ −x

4. Write the equation of the parabola created by applying each set of transformations to the graph of y = x2. a) reflection in the x-axis, vertical compression

by a factor of 0.25, translation 3 units up b) vertical stretch by a factor of 2, translation

2 units right and 4 units down c) reflection in the x-axis, vertical stretch by

a factor of 3.5, translation 3 units left and 2 units up

5. Which graph is the graph of the quadratic

relation y = 21

− (x + 3)2 + 5? Explain your

reasoning.

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a)

b)

c)

d)

6. The height, h, in kilometres, of a space tourism rocket during its unpowered flight phase is given by h = –0.005(t – 150)2 + 200, where t is the time in seconds after its rocket motor is shut down. a) What is the maximum height of the rocket? b) How long after shutdown does the rocket

reach its maximum height?


Lesson 5.3 Extra Practice Answers 1. a) vertical stretch by a factor of 2, translation

1 unit up b) translation 3 units right

c) vertical compression by a factor of 21 ,

translation 4 units down d) vertical stretch by a factor of 3, translation

3 units left and 6 units down

2. a)

b)

c)

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d.

d)

3. a) (3, 2) c) (0, –5)

b) (–7, 0) d) ⎟⎠⎞

⎜⎝⎛ −

43,

41

4. a) y = –0.25x2 + 3 b) y = 2(x – 2)2 – 4 c) y = –3.5(x + 3)2 + 2

5. The graph of y = 21

− (x + 3)2 + 5 is created from

the graph of y = x2 by a reflection in the x-axis, a

vertical compression by a factor of 21 , and a

translation 3 units left and 5 units up. These transformations produce graph d).

6. a) 200 km b) 150 s or 2 min 30 s


Chapter 5 Mid-Chapter Review Extra Practice STUDENT BOOK PAGES 273–274

1. Write the equation of each graph. a)

b)

2. Which of the following relations have parabolas that are stretches of the graph of y = x2? a) y = 1.25x2 c) y = –x2 b) y = 0.25x2 d) y = –3x2

3. Write the equation of the parabola created by applying each set of transformations to the graph of y = x2.

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d.

a) reflection in the x-axis, vertical stretch by a factor of 1.5

b) vertical stretch by a factor of 3, translation 3 units down and 4 units left

c) reflection in the x-axis, vertical compression

by a factor of 31 , and translation 5 units up

4. Sketch the graph of each quadratic relation in question 3.

5. For each pair of values of h and k, describe the transformation(s) of y = x2 that would create the graph of y = (x – h)2 + k. a) h = –3, k = 0 c) h = –1, k = 3 b) h = 2, k = 5 d) h = 0, k = –2

6. For each pair of values of h and k in question 5, sketch the graph of y = (x – h)2 + k.

7. Describe the transformations you would apply to the graph of y = x2, in the order you would apply them, to create the graph of each relation. a) y = 3x2 + 2 c) y = –(x + 3)2 + 4

b) y = 21 (x + 2)2 – 7 d) y = –2(x – 1)2 – 4

8. Which equation represents the graph shown? Explain your reasoning.

a) y = (x + 3)2 – 4 c) y = 2(x + 3)2 – 4 b) y = 2(x – 3)2 – 4 d) y = –2(x – 3)2 – 4

9. Write the equation of the parabola that matches each description. a) The graph of y = x2 is stretched vertically by

a factor of 3 and then translated 3 units down.

b) The graph of y = x2 is compressed vertically

by a factor of 51 and then translated 3 units

right and 1 unit down. c) The graph of y = x2 is reflected in the x-axis

and then translated 2 units left.

Chapter 5 Mid-Chapter Review Extra Practice | 7

Chapter 5 Mid-Chapter Review Extra Practice Answers 1. a) y = –3x2 b) y = 4x2

2. a) and d)

3. a) y = –1.5x2 b) y = 3(x + 4)2 – 3

c) y = 31

− x2 + 5

4. a)

b)

c)

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d.

5. a) translation 3 units left b) translation 2 units right and 5 units up c) translation 1 unit left and 3 units up d) translation 2 units down

6. a)

b)

c)

d)

7. a) vertical stretch by a factor of 3, translation 2 units up

b) vertical compression by a factor of 0.5, translation 2 units left and 7 units down

c) reflection in the x-axis, translation 3 units left and 4 units up

d) reflection in the x-axis, vertical stretch by a factor of 2, translation 1 unit right and 4 units down

8. The graph is created from the graph of y = x2 with a vertical stretch by a factor of 2 and a translation 3 units left and 4 units down. These transformations produce the graph in part c).

9. a) y = 3x2 – 3 c) y = –(x + 2)2

b) y = 51 (x – 3)2 – 1

8 | Principles of Mathematics 10: Chapter 5 Mid-Chapter Review Extra Practice Answers


1. a) Write an equation to describe all possible parabolas with vertex (–2, 9).

b) A parabola with vertex (–2, 9) passes through point (4, –3). Determine the value of a for this parabola.

c) Write the equation of the parabola described for part b).

d) What transformations must be applied to the graph of y = x2 to obtain the parabola described for part b)?

e) Graph the parabola described for part b).

2. Write the equation of each parabola in vertex form. a)

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d.

b)

3. Determine the equation of the parabola created by applying each set of transformations to the graph of y = x2. a) reflection in the x-axis, translation 4 units

right b) vertical stretch by a factor of 2, translation

3 units left and 2 units down

c) vertical compression by a factor of 51 ,

translation 1 unit left and 4 units up

4. Write an equation of a parabola with each set of properties. a) vertex at (–3, 2), opens upward, narrower

than y = x2 b) vertex at (2, 0), opens downward, wider than

y = x2 c) equation of the axis of symmetry x = –2,

opens upward, two zeros, same shape as y = x2

d) vertex at (3, 5), passes through (1, –3)

5. Each table of values represents a parabola. Determine the vertex of the parabola, and write the equation of the parabola in vertex form.

a) x y b) x y

–2 –11 –3 –8.5

–1 –4 –2 –9.0

0 1 –1 –8.5

1 4 0 –7.0

2 5 1 –4.5

3 4 2 –1.0

6. A golfer drives from an elevated tee. The following table gives partial data for the path of the golf ball through the air.

Time (s) 0.0 0.5 1.0 1.5 2.0 2.5

Height (m) 2.5 4.2 6.2 7.6 8.3 7.1

a) Use the data to create a scatter plot, and draw a quadratic curve of good fit.

b) Determine an equation in vertex form to model this relation.

c) Use your model for part a) to predict what happens after 4 s.

d) Check the accuracy of your model using quadratic regression.


Lesson 5.4 Extra Practice Answers 1. a) y = a(x + 2)2 + 9

b) a = 31

−

c) y = 31

− (x + 2)2 + 9

d) reflection in the x-axis, vertical compression

by a factor of 31 , translation 2 units left and

9 units up e)

2. a) y = 2x2 – 5

b) y = 21 (x + 3)2

3. a) y = –(x – 4)2 b) y = 2(x + 3)2 – 2

c) y = 51 (x + 1)2 + 4

4. a) Answers may vary, e.g., y = 3(x + 3)2 + 2 b) Answers may vary, e.g., y = –0.5(x – 2)2 c) Answers may vary, e.g., y = (x + 2)2 – 4 d) y = –2(x – 3)2 + 5

5. a) vertex at (2, 5); y = –(x – 2)2 + 5

b) vertex at (–2, –9); y = 21 (x + 2)2 – 9

6. Answers may vary, e.g., a)

b) h = –1.6(t – 2)2 + 8.3 c) The golf ball hits the ground after 4 s. d) Quadratic regression gives

h = –1.41t2 + 5.62t + 2.19. The equation in part b) is h = –1.6t2 + 6.4t + 1.9 in standard form, so the model is reasonably accurate.

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1. Determine the maximum or minimum value of a quadratic relation of the form y = a(x – h)2 + k, given the information below. a) a = –1, vertex at (3, –2)

b) a = 31 , vertex at (–2, 0)

c) a = 3, vertex at (0, 4) d) a = –0.25, vertex at (2, –5)

2. Determine the equation of a quadratic relation in vertex form, given the information below. a) vertex at (0, 2), passes through (2, 4) b) vertex at (–2, 5), passes through (0, –3) c) vertex at (7, 4), passes through (4, 13)

3. Determine the equation of each parabola in vertex form. a)

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d.

b)

c)

4. Write each equation for question 3 in standard form and factored form.

5. A quadratic relation has zeros at x = –1 and x = 3, and a minimum value of –12. Determine its equation in vertex form.

6. Express each equation in factored form and vertex form. a) y = 2x2 + 12x + 16 c) y = –4x2 – 40x – 64

b) y = 31

− x2 + 2x d) y = 2x2 – x – 1

7. The graph of y = –2(x – 3)2 + 5 was translated so that its new zeros are at 0 and 4. Determine the translation that was applied to the original graph.

8. A canal is 112 m wide. The underside of a bridge across the canal is a parabolic arch, with its bases set back 4 m from each bank of the canal. The maximum height of the arch must be at least 32 m above the surface of the canal. Write an equation to represent an arch that satisfies these conditions. Use graphing technology to graph your equation.

9. A parabola has a y-intercept of 2 and passes through points (–2, –4) and (8, –14). Determine the vertex of the parabola.


Lesson 5.5 Extra Practice Answers 1. a) maximum: –2 c) minimum: 4

b) minimum: 0 d) maximum: –5

2. a) y = 21 x2 + 2

b) y = –2(x + 2)2 + 5 c) y = (x – 7)2 + 4

3. a) y = –0.5(x – 3)2 b) y = 2(x + 3)2 – 8 c) y = 3x2 – 12

4. a) y = –0.5x2 + 3x – 4.5; y = –0.5(x – 3)2 b) y = 2x2 + 12x + 10; y = 2(x + 1)(x + 5) c) y = 3x2 – 12; y = 3(x – 2)(x + 2)

5. y = 3(x – 1)2 – 12

6. a) y = 2(x + 2)(x + 4); y = 2(x + 3)2 – 2

b) y = 31

− x(x – 6); y = 31

− (x – 3)2 + 3

c) y = –4(x + 2)(x + 8); y = –4(x + 5)2 + 36

d) y = (2x + 1)(x – 1); y = 89

412

2

−⎟⎠⎞

⎜⎝⎛ −x

7. 1 unit left and 3 units up

8. Answers may vary, e.g., y = –0.01x2, or 2

2252 xy ⎟

⎠⎞

⎜⎝⎛−=

9. (2, 4)

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1. Determine the axis of symmetry for a parabola that passes through (3, –1) and (–5, –1).

2. Use partial factoring to determine the vertex form of the quadratic relation y = x2 – 4x + 6.

3. For each quadratic relation, i) use partial factoring to determine two points

that are the same distance from the axis of symmetry

ii) determine the coordinates of the vertex iii) express the relation in vertex form iv) sketch the graph of the relation a) y = x2 – 8x + 13 c) y = –2x2 + 12x – 9

b) y = 21 x2 + x + 3 d) y = 3x2 – 9x – 2

4. Determine the axis of symmetry of the parabola y = 3x2 – 6x – 9 using each strategy. Then describe what you did. a) partial factoring b) factoring completely, if possible; if not

possible, explain why not

5. Write each relation in vertex form. a) y = (x – 3)(x + 1) c) y = 2x(x + 4) – 5 b) y = x2 – 4x d) y = x2 – 5x + 13

6. Determine the values of a and b in the relation y = ax2 + bx – 6 if the vertex of its graph is at (–2, 6).

7. Acme Inc. produces a patent line of solar batteries. Acme’s monthly profit, P, in thousands of dollars, is modelled by P = –40x2 + 320x – 420, where x is the price, in dollars, of each solar battery. What is Acme’s maximum monthly profit? What price should Acme charge to obtain this maximum profit?

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Lesson 5.6 Extra Practice Answers 1. x = –1

2. y = (x – 2)2 + 2

3. a) i) (0, 13), (8, 13) ii) (4, –3) iii) y = (x – 4)2 – 3 iv)

b) i) (0, 3), (–2, 3)

ii) ⎟⎠⎞

⎜⎝⎛−

25,1

iii) y = 21 (x + 1)2 +

25

iv)

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d.

c) i) (0, –9), (6, –9) ii) (3, 9) iii) y = –2(x – 3)2 + 9 iv)

d) i) (0, –2), (3, –2) ii) (1.5, –8.75) iii) y = 3(x – 1.5)2 – 8.75 iv)

4. a) Use partial factoring to rewrite the relation: y = 3(x2 – 2x) – 9 = 3x(x – 2) – 9. Two points on the graph are (0, –9) and (2, –9). Since these points are the same distance from the axis of symmetry, the equation of the axis of

symmetry is x = 2

20+ = 1.

b) Factor completely to rewrite the equation: y = 3(x2 – 2x – 3) = 3(x – 3)(x + 1). The relation has zeros at 3 and –1. Since the zeros are the same distance from the axis of symmetry, the equation of the axis of

symmetry is x = 2

)1(3 −+ = 1.

5. a) y = (x – 1)2 – 4 c) y = 2(x + 2)2 – 13 b) y = (x – 2)2 – 4 d) y = (x – 2.5)2 + 6.75

6. a = –3, b = –12

7. maximum profit: $220 000; price to obtain maximum profit: $4


Chapter 5 Review Extra Practice STUDENT BOOK PAGES 303–305

1. State whether each parabola is a vertical stretch or compression (or neither) of the graph of y = x2. Also state whether there is a reflection. a) y = 2x2 c) y = –x2 b) y = 0.25x2 d) y = –3.5x2

2. Sketch each parabola for question 1.

3. Describe the transformations that are applied to the graph of y = x2 to obtain each parabola. a) y = (x – 1)2 c) y = (x – 3)2 – 4 b) y = x2 + 7 d) y = (x + 1)2 – 5

4. Sketch the graph of each quadratic relation. Start with a sketch of y = x2, and apply the necessary transformations in the correct order. a) y = 3x2 + 3 c) y = –(x + 1)2 – 2 b) y = 0.25(x – 3)2 d) y = –2(x + 4)2 + 7

5. Write the equation of each parabola in vertex form. a)

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d.

b)

c)

6. Determine the equation of a quadratic relation in vertex form, given the information below. a) vertex at (3, 2), passes through (4, 4) b) vertex at (–2, 7), passes through (1, 4)

7. Express each relation in factored form and vertex form. Use partial factoring if complete factoring is not possible. a) y = 2x2 + 16x c) y = –x2 + 8x + 2 b) y = 3x2 – 12 d) y = 5x2 – 30x + 25

8. A boat tour company charges $11 for a harbour tour and averages 450 passengers on Saturdays. Over the past few months, the company has been experimenting with the price of a tour and has noticed that every increase of $1 in the price decreases the number of customers by 25. Use an algebraic model to determine the price that maximizes revenue.

9. A parabola passes through points (–1, 2), (5, 2), and (1, 4). a) Determine the coordinates of the vertex, and

write the equation of the parabola in vertex form.

b) Write the equation of the parabola in standard form.

10. Determine the values of a and c in the relation y = ax2 + 30x + c if the vertex of its graph is at (–5, 3).

Chapter 5 Review Extra Practice | 15

Chapter 5 Review Extra Practice Answers 1. a) stretch, no reflection

b) compression, no reflection c) no stretch or compression, reflection in the

x-axis d) stretch, reflection in the x-axis

2. a)

b)

c)

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d.

d)

3. a) translation 1 unit right b) translation 7 units up c) translation 3 units right and 4 units down d) translation 1 unit left and 5 units down

4. a)

b)

c)

d)

5. a) y = (x – 3)2 – 3 b) y = 0.5(x + 4)2

c) y = –2(x – 1)2 + 1

6. a) y = 2(x – 3)2 + 2 b) y = 31

− (x + 2)2 + 7

7. a) y = 2x(x + 8); y = 2(x + 4)2 – 32 b) y = 3(x – 2)(x + 2); y = 3x2 – 12 c) y = –x(x – 8) + 2; y = –(x – 4)2 + 18 d) y = 5(x – 1)(x – 5); y = 5(x – 3)2 – 20

8. $14.50

9. a) (2, 4.25); y = –0.25(x – 2)2 + 4.25 b) y = –0.25x2 + x + 3.25

10. a = 3, c = 7

16 | Principles of Mathematics 10: Chapter 5 Review Extra Practice Answers

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Chapter 5 Diagnostic Test - Nelson