chapter 1 prerequisite skillspjk.scripts.mit.edu/lab/mhf/blm_ch1_problems_all.pdf · (–3. x) d) f...

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Chapter 1 Prerequisite Skills …BLM 1–1. . Function Notation 1. Determine each value for the function

f(x) = –5x – 9. a) f(2) b) f(– 4) c) f(–3x) d) f(m + 1) 2. Determine each value for the function

f(x) = –x2 + 4x – 3. a) f(–1) b) f(0) c) f(–2x) d) 2f(m – 1) Slope and y-intercept of a Line 3. State the slope and the y-intercept of each

line a) y = – 4x – 1 b) 3y = 9x + 2 c) 2x – 4y + 12 = 0 d) x + 2y = 6(x – 1) Equation of a Line 4. Determine an equation for the line that

satisfies the following conditions.

a) The slope is 52 and the y-intercept is –2.

b) The slope is 31 and the line passes

through the point (6, –2). c) The line passes through the points

(–3, 4) and (2, –1).

Finite Differences 5. Use finite differences to determine if

each function is linear, quadratic, or neither.

a) x y −2 0 −1 −4

0 −6 1 −6 2 −4 3 0

b) x y −3 −28 −2 0 −1 10

0 8 1 0 2 −8

Domain and Range 6. State the domain and range of each

function. a) y = –3(x – 2)2 – 1 b) y = 43 −− x Quadratic Functions 7. Determine the equation of a quadratic

function that has y-intercept 4 and x-intercepts –2 and 3.

8. Determine the x-intercepts, the vertex,

the direction of opening, and the domain and range of each quadratic function. Then, graph each function.

a) y = 3(x – 1)2 – 3 b) y = (x – 2)(x + 4) c) y = –0.5(x + 1)(x – 5)

Transformations 9. Describe each transformation that must

be applied to the function y = f(x). a) y = 2f(x – 5) b) y = f(x + 1) – 4 c) y = –f(3x) + 1 10. i) Write an equation for the transformed

function of each base function. ii) Sketch a graph of each function. iii) State the domain and range of each

function. a) f(x) = x is compressed vertically by a

factor of 31 , reflected in the x-axis and

translated 4 units to the right. b) f(x) = x2 is stretched vertically by a

factor of 3, compressed horizontally

by a factor of 21 and translated 6 units

to the left and 5 units down.

11. Describe the transformation that must be applied to the base function y = x2 to obtain the graph of the function

y = – ⎟⎠⎞

⎜⎝⎛ − 4

21 x +3.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 1–1 Prerequisite Skills

Name: _______________________________ Date: ________________________

Chapter 1 Review …BLM 1–18. . (page 1) 1.1 Power Functions 1. State the degree and the leading

coefficient of each polynomial function. a) f(x) = 2x3 + 3x – 1 b) g(x) = 5x – 6 c) h(x) = x3 – 2x2 – 5x4 + 3 d) p(x) = –3x5 + 2x3 – x – 1 e) r(x) = 21 – 2x + 4x2 – 6x3

2. For each graph

i) state whether the corresponding function has even degree or odd degree

ii) state whether the leading coefficient is positive or negative

iii) state the domain and range iv) identify any symmetry v) describe the end behaviour a) Window: x∈[–4, 4], y∈[–25, 25], Yscl = 5

b) Window: x∈[–4, 4], y∈[–10, 10],

3. Complete the table. Write each function

in the appropriate row of column 2. Give reasons for your choices.

y = 3x7, y = –21 x3, y = –0.25x6, y = 2x4

End Behaviour Function Reasons

Extends from quadrant 3 to quadrant 1




1.2 Characteristics of Polynomial Function 4. Match each graph of a polynomial

function with the corresponding equation. Justify your choice.

i) y = 3x2 + 4x – 2x4 + 5 ii) y = –x5 + 3x4 + 7x3 – 15x2 – 18x a) Window: x∈[–5, 5], y∈[–50, 50], Yscl = 5

b) Window: x∈[–5, 5], y∈[–5, 15]

5. For each polynomial function in question 4 a) determine which finite differences are

constant b) find the value of the constant finite

differences 6. State the degree of the polynomial function

that corresponds to each constant finite difference and determine the value of the leading coefficient of each.

a) third differences = – 4 b) first differences = 6 c) sixth differences = –720 d) fourth differences = 96 e) second differences = –12 7. The table represents a polynomial

function. x y −3 168 −2 0 −1 −40

0 −24 1 0 2 8 3 0 4 0

Use finite differences to determine a) the degree b) the sign of the leading coefficient c) the value of the leading coefficient

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 1–18 Chapter 1 Review

Name: _______________________________ Date: ________________________

1.3 Equations and Graphs of Polynomial Functions 8. Use each graph of a polynomial function

to determine i) the x-intercept(s) and the factors of the

function ii) the least possible degree and sign of

the leading coefficient iii) the interval(s) where the function is

positive and the interval(s) where the function is negative

a) Window: x∈[–4, 6], y∈[–5, 15]

b) Window: x∈[–4, 4], y∈[–10, 60], Yscl = 5

9. Sketch a graph of each polynomial

function. a) y = 2x(x + 3)(x – 4) b) y = –3(x – 2)(x + 4)(x2 – 1)

10. The zeros of a function are –4, –52 , and

3. Determine an equation for the function if it has y-intercept 8.

11. Determine algebraically, if each

polynomial function has line symmetry about the y-axis, point symmetry about the origin, or neither. Graph the functions to verify your answer.

a) f(x) = 3x5 – 2x3 + x b) g(x) = 2x4 + 3x3 – 2x – 6 c) h(x) = 2x6 – 5x4 + x2 + 4 1.4 Transformations 12. i) Describe the transformations that must

be applied to the graph of each power function, f(x), to obtain the transformed function. Then, write the corresponding equation.

…BLM 1–18. . (page 2)

ii) State the domain and range of each transformed function.

a) f(x) = x4, y = –2f(x – 1) + 4

b) f(x) = x3, y = 31 f(2x + 6) – 5

13. Write an equation for the function that is

the result of each set of transformations. a) f(x) = x5 is stretched vertically by a

factor of 5, compressed horizontally

by a factor of 41 and translated 2 units

to the left and 1 unit down. b) f(x) = x6 is compressed vertically by a

factor of 21 , reflected in the y-axis and

translated 4 units to the right and 3 units up.

1.5 Slopes of Secants and Average Rate of Change 14. The population of a small town, p, is

modelled by the function p(t) = 10 050 + 225t – 20t2, where t is the time in years from now.

a) Determine the average rate of change of the population from

i) year 0 to year 5 ii) year 5 to year 8 iii) year 8 to year 10 b) Interpret your answers in part a). c) Graph the function to verify your

interpretation in part b). 1.6 Slopes of Tangents and Instantaneous Rate of Change 15. After being built, a car must be painted.

The revenue, R, in dollars, when x cars are painted can be modelled by the function R(x) = 1000x – 0.01x2. a) Determine the average rate of change

of revenue when painting 20 to 50 cars.

b) Estimate the instantaneous rate of change of revenue after painting 50 cars.

c) Interpret the results found in parts a) and b).

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 1–18 Chapter 1 Review

Name: _______________________________ Date: ________________________

1.1 Power Functions …BLM 1–3. . 1. State the degree and the leading

coefficient of each polynomial. a) y = –2x5 + 3x3 – 1 b) y = 3x + 5x2

c) y = 3

4x – 3x3 + x – 6

2. Complete the table for the following

functions. y = 3x y = –2x7 y = –3x4

y = 4x5 y = –x2 y = 32 x6

y = –61 x3 y = 2x10

End Behaviour Function Reasons





3. Consider each graph. i) Does it represent a power function, an

exponential function, a periodic function, or none of these?

ii) Identify any symmetry. a) b)

c) d)

e) f)

4. a) Graph y = x4, y = –3x4, and y = 21 x4 on

the same set of axes. b) Compare and describe the key features

of the graphs. 5. a) Graph y = x5, y = (x + 3)5, and

y = (x – 3)5 on the same set of axes. b) Describe the similarities and differences

among the graphs in part a). c) Make a conjecture about the

relationship between the graphs of y = xn and y = (x – h)n, where h∈ and n is an odd whole number.

6. The surface area of a spherical snowball

is given by the function S(r) = where r is the radius of the snowball, in centimetres, and r

24πr ,

∈[0, 12]. a) Graph S(r). b) State the domain and the range. c) Describe the similarities and

differences between the graphs of S(r) and y = x2.

7. a) Describe the relationship between the

graph of y = x3 and the graph of

y = –21 (x + 1) 3– 4

b) Predict the relationship between the graph of y = x5 and the graph of

y = –21 (x + 1) 5– 4

c) Verify the accuracy of your prediction by graphing the functions in part b).

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 1–3 Section 1.1 Practice

Name: _______________________________ Date: ________________________

1.2 Characteristics of Polynomial Functions …BLM 1–8. . (page 1) 1. Each graph represents a polynomial

function of degree 3, 4, 5 or 6. Determine the least possible degree of the function corresponding to each graph. Justify your answer.

a) Window: x∈[–4, 4], y∈[–10, 10]

b) Window: x∈[–4, 4], y∈[–10, 10]

c) Window: x∈[–4, 4], y∈[–10, 10]

d) Window: x∈[–4, 4], y∈[–12, 12]

2. Refer to question 1. For each graph do

the following. i) State the sign of the leading

coefficient. Justify your answer. ii) Identify any symmetry.

3. Use the degree and the sign of the leading coefficient to i) describe the end behaviour of each

polynomial function ii) state which finite differences will be

constant iii) determine the value of the constant

finite differences a) f(x) = 2x3 – 4x2 + x – 3 b) f(x) = –3x2 – 4x + 1 c) f(x) = –x4 + 2x2 + 2 d) f(x) = 2x + 6 e) f(x) = x5– 3x3 +2x + 4

4. State the degree of the polynomial

function that corresponds to each constant finite difference. Determine the value of the leading coefficient for each polynomial function.

a) third differences = 24 b) fifth differences = –240 c) second differences = 12 d) fourth differences = –96 5. Consider the function

g(x) = –2x4 + 3x2 + 6x – 1. a) Without graphing, determine

i) the end behaviour of the function ii) which finite differences will be

constant iii) the value of the constant finite

differences b) Sketch a graph of the polynomial

function. 6. Each table represents a polynomial

function. Use finite differences to determine the following for each

i) the degree ii) the value of the leading coefficient

a) x y −2 −3 −1 2

0 1 1 6 2 29 3 82 4 177


Name: _______________________________ Date: ________________________

b)

x y −1 −13

0 −4 1 −7 2 −22 3 −49 4 −88 5 −139

7. An open topped rectangular box has a

volume, V, in cubic centimetres, that can be modelled by the function V(x) = x(x – 8)(x – 20), where x is the length of the base of the box, in centimetres. a) Without calculating, determine which

finite differences are constant for this polynomial function and the value of that constant. Justify your answer.

b) Describe the end behaviour of the function, assuming that there are no restrictions on the domain.

c) Graph V(x). State the restrictions on the domain in this situation.

…BLM 1–8. . (page 2) 8. Graph a polynomial function that

satisfies each description. a) a cubic function with a negative

leading coefficient and one x-intercept b) a quintic function with a positive

leading coefficient and 3 x-intercepts c) a quadratic function with a negative

leading coefficient and no x-intercepts d) a quartic function with a positive

leading coefficient and 4 x-intercepts


Name: _______________________________ Date: ________________________

1.3 Equations and Graphs of Polynomial Functions …BLM 1–11. . (page 1) 1. For each polynomial function

i) state the degree and sign of the leading coefficient

ii) describe the end behaviour of the graph of the function

iii) determine the x-intercepts a) f(x) = (x + 2)(3 – x)(x + 1) b) p(x) = –2x(x – 4)2(2 – x) c) h(x) = x(1 – 2x)3(x + 3)2 d) f(x) = (2x – 5)2(x – 2)2(x + 1)

2. For each graph do the following.

i) State the x-intercepts. ii) State the least possible degree and the

sign of the leading coefficient. iii) Give the interval(s) where the

function is positive and the interval(s) where the function is negative.

a)

b)

c) Window: x∈[–6, 6], y∈[–5, 15]

3. i) Determine the zeros of each

polynomial function. Indicate whether the zeros are of order 1, 2, or 3.

a) f(x) = 2x(x – 2)(3 – x) b) g(x) = (x + 1)3(5x – 2)2 c) h(x) = 3(x + 2)2

d) p(x) = –(x + 3)2(x – 3)2 ii) Determine algebraically if each

function is even, odd, or neither. iii) Sketch a graph of each function.

4. i) Determine algebraically whether each

function has point symmetry about the origin or line symmetry about the y-axis. State whether each function is even, odd, or neither. Explain.

a) f(x) = –4x(x – 1)(x + 1) b) g(x) = (x – 2)3(x + 2)3

c) h(x) = 2x(x – 3)(x + 4) d) p(x) = –(x – 4)(x – 2)(x + 4)(x + 2) e) y = 3x6 – 2x4 + 1 f) y = –5x3 – 3x ii) Verify your answers in part i) using

technology. 5. Determine an equation for the polynomial

function that corresponds to each graph. a) Window: x∈[–5, 5], y∈[–5, 20], Yscl = 2

b) Window: x∈[–5, 5], y∈[–10, 30], Yscl = 5

c)


Name: _______________________________ Date: ________________________

6. Determine an equation for each

polynomial function. State whether each function is even, odd, or neither. Sketch a graph of each function. a) a quartic function with zeros –1,

3(order 2), and 5 and y-intercept 3

b) a cubic function with zeros –3, –21 ,

and 4 and y-intercept 24

c) a quintic function with zeros –32 ,

–2(order 2), and 3(order 2) and y-intercept –36

7. Each polynomial function has zeros at

–21 , 3, and 5. Write an equation for each

function. Then, sketch a graph of the function. a) a quartic function with a negative

leading coefficient b) a quintic function that touches the

x-axis at –21 and at 5

c) a cubic function that extends from quadrant 2 to quadrant 4.

…BLM 1–11. . (page 2) 8. i) Determine the zeros of each

polynomial function. a) f(x) = x3 – 27 b) g(x) = (x2 – 64)(x2 + 3x – 4) c) h(x) = 2x3 – 3x2 – 2x + 3

ii) Use graphing technology to verify your answers in part i).

9. a) Write an equation for an odd function

with three x-intercepts, one of which

is 27 .

b) Verify algebraically that you have created an odd function in part a).

c) Determine an equation for a function that has the three x-intercepts from part a) and that passes through the point (1, –15).


Name: _______________________________ Date: ________________________

1.4 Transformations …BLM 1–14. . (page 1) 1. a) The graph of y = x3 is transformed to

obtain the graph of y = –2(3x)3 + 5. State the parameters and describe the corresponding transformations.

b) Complete the table. y = x3 y = (3x)3 y = −2(3x)3 y = −2(3x)3 + 5

(−2, −8) (−1, 1) (0, 0) (1, 1) (2, 8)

c) Sketch a graph of y = –2(3x)3 + 5. 2. Compare each polynomial function with

the equation y = a[k(x – d)]n + c. State the values of the parameters a, k, d and c and the degree n, assuming that the base function is a power function. Describe the transformation that corresponds to each parameter.

a) y = 21 (x + 3)2 – 5

b) y = [2(x − 2)]3 + 0.5 c) y = –(3x – 6)4 – 2 d) y = 4(–x – 6)3 + 1 3. Match each graph with the corresponding

function. Justify your choice.

i) y = – 21 x4 + 1 ii) y =

3

121

⎟⎠⎞

⎜⎝⎛ +x

iii) y = 21 (x – 1)3 iv) y =

21 (x – 1)4

a) Window: x∈[–5, 5], y∈[–10, 10]

b) Window: x∈[–5, 5], y∈[–10, 10]

c) Window: x∈[–5, 5], y∈[–10, 10]

d) Window: x∈[–5, 5], y∈[–10, 10]

4. Describe the transformation that must be

applied top the graph of each power function f(x) to obtain the transformed function. Write the transformed function.

a) f(x) = x3, y = 2f(x – 3) – 4 b) f(x) = x4, y = –f(2x + 6) + 1 5. a) Given a base function of y = x4, list the

parameters of the polynomial function

y = 21 [3(x – 1)]4 + 2.

b) Describe how each parameter in part a) transforms the graph of y = x4.

c) Determine the domain, range, vertex and equation of the axis of symmetry for the transformed function.


Name: _______________________________ Date: ________________________

6. i) Describe the transformation that must

be applied to the graph of each power function, f(x), to obtain the transformed function.

a) f(x) = x2, y = –3f(x – 2) – 4 b) f(x) = x4, y = 2f(2x + 6)

c) f(x) = x3, 1 1 ( 1) 52 2

y f x⎛ ⎞= − − −⎜ ⎟⎝ ⎠

ii) Write the full equation of each transformed function in part i).

iii) Sketch each base function from part i) and its transformed function from part ii) on the same set of axes.

iv) State the domain and range of each pair of functions in part iii).

7. Transformations are applied to each

power function to obtain the resulting graph. Determine an equation for the transformed function. All graphs have window variables x∈[–6, 6], y∈[–5, 10]. a)

b)

c)

…BLM 1–14. . (page 2) 8. i) Write an equation for the function that

results from the given transformations. ii) State the domain and range for each

function. For even functions, give the vertex and the equation of the axis of symmetry.

a) The function f(x) = x5 is compressed

vertically by a factor of 21 and

translated 3 units down. b) The function f(x) = x3 is stretched

vertically by a factor of 2, reflected in the x-axis and translated 4 units to the right and 1 unit up.

c) The function f(x) = x4 is reflected in the x-axis, stretched horizontally by a factor of 3 and translated 5 units to the left.

d) The function f(x) = x4 is stretched vertically by a factor of 3, stretched horizontally by a factor of 2, reflected in the y-axis and translated 1 unit to the right and 6 units down.

9. a) Predict the relationship between the

graph of y = x4 + x3 and the graph of y = [(x + 3)4 + (x + 3)3] – 1.

b) Use Technology Graph each function in part a) to verify the accuracy of your prediction.

c) Determine the x-intercepts of each function in part a). Round your answers to 1 decimal place.

d) Give the approximate domain and range of each function in part a). Round your answers to one decimal place.


Name: _______________________________ Date: ________________________

1.5 Slopes of Secants and Average Rate of Change …BLM 1–15. . 1. Identify whether the average rate of

change for pairs of points along each graph is constant and positive, constant and negative, zero or non-constant. Justify your response. Window variables for all three graphs: x∈[0, 10], y∈[0, 10]. a)

b)

c)

2. Determine the average rate of change for

two points on each line segment in question 1.

3. The cost, C, of making one unit of a product at any time x, in years, can be modelled by the function C(x) = 4500 + 1530x – 0.04x3, x∈[0, 200]. a) Determine algebraically the average

rate of change of the cost from i) year 0 to year 4 ii) year 4 to year 7 iii) year 7 to year 9 b) Interpret your answers from part a).

4. If a ball is dropped from the top of a

120-m cliff, its height, h, in metres, after t seconds can be modelled by h(t) = 120 – 4.9t2. a) Find the average rate of change of the

height of the ball with respect to time over the intervals

i) 1 s to 4 s ii) 4 s to 6 s iii) 6 s to 7 s b) What does the average rate of change

represent in this situation? c) Interpret the significance of your

answers in part a). 5. A census is taken of the permanent

population, P, of Collingwood, Ontario every five years.

Year Population 1991 13 500 1996 15 596 2001 16 039 2006 17 290

a) Determine the average rate of change of the population for

i) 1991 to 1996 ii) 1996 to 2001 iii) 2001 to 2006 iv) the entire fifteen-year period b) What factors do you think might have

led to the varied results?


Name: _______________________________ Date: ________________________

1.6 Slopes of Tangents and Instantaneous Rate of Change …BLM 1–17. . 1. Consider the graph shown.

Window: x∈[−6, 6], y∈[−12, 8]

a) Write the coordinates of the tangent point shown.

b) Estimate and state the coordinates of another point on the tangent line.

c) Determine the slope of the tangent using the points from parts a) and b).

d) Draw the tangent to the curve at the point (–1, –5). Estimate another point that is on this tangent line. Find the slope of this second tangent line.

e) Interpret the significance of the differences in the values found in parts c) and d).

f) Discuss what you predict the values will be of the slopes of tangents drawn at other points on the curve.

2. A projectile is shot into the air such that

its height, h, in metres, after t seconds can be modelled by the function h(t) = 25t – 4.9t2. a) Complete the table.

Interval

hΔ

tΔ

ht

ΔΔ

4 ≤ t 4.1 ≤ 4 ≤ t ≤ 4.01

≤ t ≤ 44.001

b) e velocity

3. ,

a)

with respect to time for the first 6 h.

Use the table to estimate thof the projectile after 4 s.

After t hours, the number of bacteria, Nin a culture can be modelled using the function N(t) = 75 000 + 64t3.

Determine the average rate of change of the growth of the number of bacteria

b) Estimate the instantaneous rate of change of the number of bacteria after 6 h.

4. The displacement, s , in metres, of a

particle moving back and forth in a straight line can be modelled by the function s(t) = 4t2 – 10t + 13 where t is measured in seconds. a) Find the average rate of change of the

distance with respect to time from 1 s to 4 s.

b) Estimate the instantaneous rate of change of the displacement of the particle after 1 s.

c) Sketch the curve and the tangent at t = 1.

d) Interpret the average rate of change and the instantaneous rate of change for this situation.

5. The population, P, of a small town is

modelled by the function P(t) = –2t3 + 55t2 + 15t + 22 000, where t = 0 represents the beginning of this year. a) Write an expression for the average

rate of change of population from t = 10 to t = 10 + h.

b) Use the expression in part a) to determine the average rate of change of the population when

i) h = 3 ii) h = 5 c) Use the expression in part a) to

estimate the instantaneous rate of change of the population after 10 years.

d) Use Technology Graph P(x). e) Using the graph from part d), would it

be justified for a large department store to open 10 years from now in this town? Explain.

f) If the store was not opened10 years from now, would it be justified instead to open it 30 years from now? Explain.


Chapter 1 Practice Masters Answers …BLM 1–22. . (page 1)

Prerequisite Skills 1. a) –19 b) 11

c) 15x – 9 d) –5m – 14 2. a) –8 b) –3

c) – 4x2 – 8x – 3 d) –2m2 + 12m – 16

3. a) m = – 4, b = –1 b) m = 3, b = 32

c) m = 21 , b = 3 d) m =

25 , b = –3

4. a) y = 52 x – 2 b) y =

31 x – 4

c) y = –x + 1 5. b) neither a) quadratic 6. a) { x∈ }, { 1y ≤ − } , y∈

b) 4, 3

x x∈ ≤ − }, { }

7. y = –

, 0y y∈ ≥

32 x2 +

32 x + 4

ertex (1, –3); opens 8. a) x-intercepts 2, 0; vupward; { x∈ }; { , 3y y∈ ≥ − }

b) x-intercepts – 4, 2; vertex (–1, –9);

opens upward; { x∈ }, { , 9y y∈ ≥ − }

c) x-intercepts –1, 5; vertex (2, 4.5);

opens downward; { x∈ }; { , 4.5y y∈ ≤ }

9. a) vertical stretch by a factor of 2 and a

b) d 4 translation 5 units to the right translations 1 unit to the left anunits down

c) reflection in the x-axis, horizontal

compression by a factor of 31 and a

translation 1 unit up

10.a) i) y = –31 (x – 4)

ii)

iii) { x∈ }; { y∈ }

b) y = 3[2(x + 6)]2 – 5 ii)

iii) { x∈ }; { , 5y y∈ ≥ − }

11. reflection in the x-axis, horizontal stretch by a factor of 2 and translation 8 units to the right and 3 units up

1.1 Power Functions 1. a) degree 5; leading coefficient –2

b) degree 2; leading coefficient 5

c) degree 4; leading coefficient 31

2. quadrant 3 to 1: y = 3x, y = 4x5; reason: odd degree with positive leading

coefficient; quadrant 2 to 4: y = –61 x3, y

= –2x7; reason: odd degree with negative leading coefficient; quadrant 2 to 1: y =

32 x6, y = 2x10 reason: even degree with

positive leading coefficient; quadrant 3 to 4: y = –x2, y = –3x4; reason: even degree with negative leading coefficient

3. i) a) exponential function b) power function c) power function d) not a function e) none of the functions mentioned f) a periodic function

ii) a) symmetry: none b) line symmetry about the y-axis

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 1–22 Chapter 1 Practice Masters Answers


c) line symmetry about the y-axis d) symmetry in the x-axis e) point symmetry about (–3, 0) f) point symmetry about the origin

4. a) Window: x∈[–4, 4], y∈[–10, 10]

b) y = x4: quadrant 2 to 1, line symmetry

about the y-axis ; y = –3x4: quadrant 3 to 4, line symmetry about the y-axis;

y = 21 x4: quadrant 2 to 1, line

symmetry about the y-axis 5. a)

b) similarities: same end behaviour:

quadrant 3 to 1; same shape; same domain and range differences: y = x5: point symmetry about the origin; y = (x + 3)5: point symmetry about (–3, 0); y = (x – 3)5: point symmetry about (3, 0); different x- and y-intercepts

c) They will have the same end behaviour and shape, but different point symmetry.

6. a)

b) { };

{ } , 0 12r r∈ ≤ ≤, 0 57S S∈ ≤ ≤ 6π

c) Similarities: end behaviour to the right of the y-axis is in quadrant 1 ; Differences: S(r) is positive only, narrower than y = x2 with no symmetry; y = x2 has positive and negative values with symmetry in the y-axis.

7. a) y = –0.5(x + 1)3 – 4 is compressed vertically by a factor of 0.5, reflected in the x-axis and translated 1 unit to the left and 4 units down ; the shape is the same ; the end behaviour and symmetry differ

b) same transformations and same relationship

c) Window: x∈[–4, 4], y∈[–10, 10]

1.2 Characteristics of Polynomial Functions 1. a) least degree 3; one x-intercept, goes

from quadrant 3 to quadrant 1 b) least degree 4; two x-intercepts of

order 1 and one x-intercept order 2 and 1 local maximum, 2 local minima with a total of 3 which is one less than the degree

c) least degree 4: two x-intercepts but 2 local maxima, 1 local minimum with a total of 3 which is one less than the degree

d) least degree 5: five x-intercepts and 2 local maxima, 2 local minima with a total of 4 which is one less than the degree

2. i) a) positive – quadrant 3 to 1 b) positive – graph opens upward, quadrant 2 to 1 c) negative – graph opens downward, quadrant 3 to 4 d) negative – quadrant 2 to 4

ii) no line symmetry, a) has point symmetry

3. a) i) quadrant 3 to 1 ii) third iii) 12 b) i) quadrant 3 to 4 ii) second iii) −6 c) i) quadrant 3 to 4 ii) fourth iii) −24 d) i) quadrant 3 to 1 ii) first iii) 2 e) i) quadrant 3 to 1 ii) fifth iii) 120

4. a) 3, 4 b) 5, –2 c) 2, 6 d) 4, –4 5. a) i) quadrant 3 to 4 ii) fourth iii) – 48



b)

6. a) i) 3 ii) 2 b) i) 2 ii) –6 7. a) third, 6

b) quadrant 3 to 1 c) { } Window: x∈[0, 10],

y∈[0, 300], Yscl= 20 , 0 8x x∈ < <

8. Answers may vary. 1.3 Equations and Graphs of Polynomial Functions 1. a) i) 3; negative ii) quadrant 2 to 4

iii)–2, –1, 3 b) i) 4; positive ii) quadrant 2 to 1

iii) 0, 2, 4(order 2) c) i) 6; negative ii) quadrant 3 to 4

iii) –3(order 2), 0, 0.5(order 3) d) i) 5; positive ii) quadrant 3 to 1

iii) –1, 2(order 2), 2.5(order 2) 2. a) i) –1, 2, 4 ii) 3, positive

iii) positive −1 < x <2, x > 4; negative x < −1, 2 < x < 4

b) i) –2, 3 ii) 4, negative iii) negative x < −2, 2 < x < 3, x > 3

c) i) –3, 1.5 ii) 2, negative iii) positive –3 < x < 1.5; negative: x < –3, x > 1.5

3. i) a) 0, 2, 3 (all order 1)

b) –1 (order 3), 52 (order 2)

c) –2 (order 2) d) –3, 3 (both order 2)

ii) a) neither b) neither c) neither d) even

4. i) a) odd b) even c) neither d) even e) even f) odd

5. a) y = (x + 3)(x – 2)2 b) y = –x(x + 3)(x – 2)(x – 1) c) y = 0.5(x + 5)(x – 2)

6. a) y = –151 (x + 1)(x – 3)2(x – 5); neither

b) y = –2(x + 3)(2x + 1)(x – 4); neither

c) y = –

21 (3x + 2)(x + 2)2(x – 3)2; neither

7. Answers may vary. Sample answers:

a) y = −(2x − 1)(x − 3)2(x −5) b) y = (2x − 1)2(x − 3)(x −5)2 c) y = −(2x − 1)(x − 3)(x −5)

8. i) a) 3 b) –8, –4, 1, 8 c) –1, 1, 1.5 9. a) y = x(2x + 7)(2x – 7)

c) y = 31 x(2x + 7)(2x – 7)

1.4 Transformations 1. a) a: vertical stretch by a factor of 2 with

a reflection in the x-axis; k: horizontal

compression by a factor of 31 ; c:

translation 5 units up y = x3 y = (3x)3 y =

−2(3x)3 y = −2(3x)3 +

5 (−2, −8) (−2, −216) (−2, 432) (−2, 437) (−1, 1) (−1, −27) (−1, 54) (−1, 59) (0, 0) (0, 0) (0, 0) (0, 5) (1, 1) (1, 27) (1, −54) (1, −49) (2, 8) (2, 216 (2, −432) (2, −427)



2. a) a = 0.5: vertical compression by a factor of 0.5; k = 1: no horizontal stretch or compression; d = –3: horizontal translation 3 units to the left; c = –5: vertical translation 5 units down; n = 2

b) a = 1: no vertical stretch or compression; k = 2: horizontal

compression by a factor of 21 ; d = 2:

horizontal translation 2 units to the right; c = 0.5: vertical translation 0.5 units up; n = 3

c) a = –1: reflection in the x-axis; k = 3: horizontal compression by a factor of

31 ; d = 2: horizontal translation

2 units to the right; c = –2: vertical translation 2 units down; n = 4

d) a = 4: vertical stretch by a factor of 4; k = –1: reflection in the y-axis; d = –6: horizontal translation 6 units to the left; c = 1: vertical translation 1 unit up; n = 3

3. i) and b); ii) and d); iii) and c); iv) and a) 4. a) vertical stretch by a factor of 2 with

translations 3 units to the right and 4 units down; y = 2(x – 3)3 – 4

b) reflection in the x-axis, horizontal compression by a factor of 0.5 with translations 3 units to the left and 1 unit up; y = –(2x + 6)4 + 1

5. a) a = 0.5, k = 3, d = 1, c = 2 b) vertical compression by a factor of

0.5, a horizontal compression by a

factor of 31 and translations 1 unit to

the right and 2 units up c) { }; { }; vertex; (1,

2); axis of symmetry: x = 1 x∈ , 2y y∈ ≥

6. a) i) a vertical stretch by a factor of 3, a reflection in the x-axis and translations 2 units to the right and 4 units down ii) y = –3(x – 2)2 – 4

iii)

iv) base: {x∈ }; { } , 0y y∈ ≥

new: {x∈ }; { } , 4y y∈ ≤ −b) i) a vertical stretch by a factor of 2, a

horizontal compression by a factor of 0.5 and a translation 3 units to the left ii) y = 2(2x + 6)4 iii)

iv) both: { x∈ }; { } , 0y y∈ ≥

c) i) a vertical compression by a factor of 0.5, a reflection in the x-axis, a horizontal stretch by a factor of 2 and translations 1 unit to the right and 5 units down ii) y = –0.5[0.5(x – 1)]3 – 5 iii)

iv) both: { x∈ }; { } y∈

7. a) y = 0.5(x – 3)4 + 4 b) y = –x3 + 3 c) y = [–(x – 3)]3

8. a) i) f(x) = 0.5x5 – 3 ii) { }; {

x∈y∈ }

b) i) y = –2(x – 4)3 + 1 ii) { x∈ }; { y∈ }

c) i) y = –[31 (x + 5)]4

ii) { x∈ }; { , 0y y∈ ≤ }; (–5, 0); x = –5

d) i) y = 3[–0.5(x – 1)]4 – 6 ii) { x∈ }; { , 6y y∈ ≥ − }; (1, –6); x = 1

9. a) transformed graph is translated 3 units to the left and 1 unit down

c) original: 0, –1 ; new: – 4.4, –2.2



d) original: { } ; { }; new: { }; { }

x∈ , 0.1y y∈ ≥ −, 1.1y y∈ ≥ −x∈

1.5 Slopes and Secants and Average Rate of Change 1. a) constant and negative

b) constant and positive c) zero 2. a) –1 b) 2 c) 0 3. a) i) 1529.36 ii) 1526.28 iii) 1522.28

b) The average rate of change (the slope of the secant line) is positive over each time period, but gradually decreasing.

4. a) i) –24.5 m/s ii) – 49 iii) –63.7 b) velocity c) The answers are negative because the

ball is moving towards and getting closer to the ground and the ball falls faster as it gets closer to the ground.

5. a) i) 419 ii) 88 iii) 250 iv) 252 b) Answers will vary.

1.6 Slopes of Tangents and Instantaneous Rate of Change 1. a) (2, −8) b) (6, 0) c) m = 2 d) m = −4

e) Slopes of tangent lines drawn from any point on this curve will be positive to the right of the minimum, negative to the left of the minimum and zero at the minimum.

2. a) thΔΔ : −14.7, −14.25, −14.2 b) −14.2

3. a) 2304 bacteria per hour b) 6912 bacteria per hour

4. a) 10 m/s b) −2 m/s c) Window: x∈[0, 5], y∈[−20, 50], Yscl = 5

d) The average rate of change is the slope

of the secant line drawn from t = 1 to t = 4 which is positive and represents the average velocity between those time values. The instantaneous rate of change is the slope of the tangent line drawn at t = 1 which is negative and represents the instantaneous velocity after 1 s.

5. a) (10 ) (10)f h fh

+ − = –2h2 – 5h + 515

b) i) 482 ii) 440 c) 515 d)

e) increased population, 25 650, so store

justified f) decreased population, 17 950, store

probably not justified Chapter 1 Review 1. a) degree: 3; leading coefficient: 2

b) degree: 1; leading coefficient: 5 c) degree: 4; leading coefficient: –5 d) degree: 5; leading coefficient: –3 e) degree: 3; leading coefficient: –6

2. a) i) even degree ii) negative iii) { x∈ }; { , 15y y∈ ≤ } iv) no line or point symmetry v) quadrant 3 to 4

b) i) odd degree ii) positive iii) { x∈ }; { y∈ } iv) point symmetry about (0, 0) v) quadrant 3 to 1

3. y = 3x7: quadrant 3 to 1, odd degree with

positive leading coefficient; y = 21

− x3:

quadrant 2 to 4, odd degree with negative leading coefficient; y = 2x4: quadrant 2 to 1, even degree with positive leading coefficient; y = –0.25x6: quadrant 3 to 4, even degree with negative leading coefficient

4. i) and b); ii) and a) 5. a) i) fourth ii) fifth

b) i) – 48 ii) –120

6. a) 3,32

− b) 1, 6 c) 6, –1

d) 4, 4 e) 2, –6 7. a) fourth b) positive c) 1 8. a) i) –1, 0, 4; x(x + 1)(x − 4)

ii) 3, negative iii) positive x < −1, 0 < x < 4;

negative −1 < x < 0, x > 4



b) i) –3, –2, 3; (x + 3)(x + 2)(x – 3)2 ii) 4, positive iii) positive x < −3, –2 < x < 3, x > 3;

negative –3 < x < –2

10. y = 31

− (x + 4)(5x + 2)(x – 3)

11. a) point symmetry about the origin b) neither c) line symmetry about the y-axis

12. a) i) a vertical stretch by a factor of 2, a reflection in the x-axis and translations 1 unit to the right and 4 units up; y = −2(x − 1)4 + 4 ii) { , { } , 4}y y∈ ≤x∈

b) i) a vertical compression by a factor of

31 , a horizontal compression by a

factor of 21 and translations 3 units to

the left and

5 units down; 31 (2 6) 53

y x= + −

ii) { } , { x∈ }y∈13. a) y = 5[4(x + 2)]5 – 1

b) y = 0.5[–(x − 4)]6 + 3 14. a) i) 125 ii) –35 iii) –135

b) The population grew in the first 5 years towards its maximum; then it began to decrease in the next 3 years and decreased considerably in the following 2 years.

c) Window: x∈[0, 10], y∈[0, 13 000], Yscl = 1000

15. a) $999.30 b) $999.00

c) The slope of the secant line from x = 20 to x = 50 approaches the slope of the value of the tangent line at x = 50, i.e. the average rate of change from x = 20 to x = 50 is quite close in value to the instantaneous rate of change at x = 50.

Chapter 1 Test 1. B

2. D 3. i) and c): odd degree, negative leading

coefficient; ii) and b): even degree, positive leading coefficient; iii) and a): odd degree, positive leading coefficient

4. i) fifth; –120 ii) fourth; 24 iii) third; 6 5. a) y = a(x + 4)(x + 2)(x – 1)2, a ≠ 0

b) y = 43

− (x + 4)(x + 2)(x – 1)2

c)

positive −4 < x < −2; negative x < −4, −2 < x < −1, x > 1

6. a) a = 4: a vertical stretch by a factor of 4; k = 3: a horizontal compression by a

factor of 31 ; d = 4: a horizontal

translation 4 units to the right; c = –2: a vertical translation 2 units down

b) { }x∈ ; { , 2}y y∈ ≥ − ; vertex: (4, –2); axis of symmetry: x = 4

c)

7. y = –(x + 2)4 + 4 8. a) cubic function with a positive leading

coefficient ; end behaviour: quadrant 3 to 1; no point or line symmetry; x-intercept –16.4; y-intercept 25 000; no local maximum or minimum points

b) 25 000 c) 36 d) t 0, P(t) 0 ≥ ≥e) 25.3 years

9. a) –20.3 m/s b) –32.5 m/s c) Both values are negative because the

stone is moving downward towards the ground.


chapter 1 prerequisite skillspjk.scripts.mit.edu/lab/mhf/blm_ch1_problems_all.pdf · (–3. x) d) f...

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