chapter 8 prerequisite skillspjk.scripts.mit.edu/lab/mhf/chapter_8_practice.pdfblm 8–3 section 8.2...

Name: _______________________________ Date: ________________________

Chapter 8 Prerequisite Skills …BLM 8–1. . Identify Linear, Quadratic, and Exponential Growth Patterns 1. Consider this pattern built from

equilateral triangles.

a) Draw the next stage of the pattern. b) Is the pattern linear, quadratic,

exponential, or other? c) Create a scatter plot of the total

number of triangles, T, in stage n of the pattern.

d) Determine the equation relating T and n.

Graph and Analyse Power Functions 2. For what values of n is ( ) nf x x= a) an even function? b) and odd function? Graph and Analyse Rational Functions

3. Sketch the graph of 12

yx

=+

, and state

its domain and range.

4. Consider the function 2

12 3

xyx x

+=

− −.

a) Simplify the function and state any restrictions on the variables.

b) State the domain and range. c) Sketch the graph. d) Identify any asymptotes or holes in the

graph.

5. Sketch a possible position-time graph that would show the height of a marble released from a position on the side of a bowl whose cross section is shown in the diagram.

Inverses 6. Determine the defining equation for the

inverse of each function. a) 2( ) ( 2) 5f x x= + +

b) 1( ) 42

f xx

= +−

7. Which inverses in question 6 are

functions? Explain.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 8–1 Prerequisite Skills

Name: _______________________________ Date: ________________________

8.1 Sums and Differences of Functions …BLM 8–2. . 1. Let ( ) 3 5f x x= − and . ( ) 2 3g x x= +

a) Write the equation for . ( ) ( ) ( )h x f x g x= +

b) Determine the value of h(2). c) Write the equation for

. ( ) ( ) ( )k x f x g x= −d) Determine the value of k(2).

2. Use the functions f(x) and g(x) as shown.

a) Apply the superposition principle to

graph and .

( ) ( )y f x g x= +( ) ( )x g x= −y f

b) Give the domain and range of . ( ) ( )y f x g x= +

c) Give the domain and range of . ( ) ( )y f x g x= −

3. Let 2( )f x x= and 1( )g xx

= .

a) Graph , , and on the same set of

axes.

( )y f x=( ) ( )x g x= +

( )y g x=y f

b) State the domain and range of . ( ) ( )y f x g x= +

c) For large values of x, does behave more like

or like ? Explain. ( ) ( )y f x g x= +( )y f x= ( )y g x=

4. A salesperson has fixed costs of

$1500 per month and variable costs of $200 per unit sold. She earns $250 per unit sold. She can sell a maximum of 200 units per month. a) Write an equation for C, her total cost,

as a function of n, the number of units sold.

b) Write an equation for R, her revenue, as a function of n, the number of units sold.

c) Graph C and R on the same set of axes. d) What is the break-even point? e) Write the equation for her profit

( ) ( ) ( )P n R n C n= − . f) In this context, what are the domain

and range for P? g) What value would she need to reduce

her variable costs to, in order to have a break-even point of 20 units sold?

5. a) Sketch graphs of ( ) sinf x x= and

( ) cosg x x= on the same set of axes. Use the domain 3π 3πx− ≤ ≤ . b) Use the principle of superposition to

sketch a graph of . ( ) ( )y f x g x= +c) Determine the equation of

( ) ( )y f x g x= + . Express your answer as a sine function.

d) Sketch a graph of . ( ) ( )y f x g x= −e) Determine the equation of

( ) ( )y f x g x= − . Express your answer as a sine function.

6. Use Technology Investigate Fourier series.

a) Graph each function.

( ) ( )1sin sin

33y x= + x

( ) ( ) ( )1 1sin sin 3 sin 5

3 5x x xy = + +

( ) ( ) ( ) ( )1 1 1sin sin 3 sin 5 sin 7

3 5 7x x xy = + + + x

b) What shape do the graphs seem to be approaching as more terms are added? Test your answer by graphing the function with two more terms added.

c) Repeat parts a) and b) for the function

( ) ( ) ( ) ( )1 1 1 1sin 2 sin 4 sin 6 sin 8

2 4 6 8y x x x x= + + + K

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 8–2 Section 8.1 Practice

Name: _______________________________ Date: ________________________

8.2 Products and Quotients of Functions …BLM 8–3. . (page 1)

1. Let ( ) 2f x = x and 1( )2

g xx

=−

.

a) Give the domain and range of each function.

b) Develop a graphical model for

and ( ) ( )y f x g x=( )( )

f xyg x

= .

c) Give the domain and range of each combined function.

d) Identify any asymptotes and holes in the combined functions

2. Let ( ) 5f x x= + and ( ) cosg x x= .

a) State the domain and range of each function.

b) Develop an algebraic model for

and ( ) ( )y f x g x=( )( )

f xyg x

= .

c) Use graphing technology to graph

and ( ) ( )y f x g x=( )( )

f xyg x

=

d) Give the domain and range of each combined function. (Estimate the range.)

e) Identify any asymptotes or holes in the combined functions.

3. Let ( )f x x= and 2( )g x x=

a) Graph and describe its shape. Is f(x) even, odd, or neither?

( )y f x=

b) Graph on the same set of axes as and describe its shape. Is g(x) even, odd, or neither?

( )y g x=( )y f= x

c) Sketch graphs of and ( ) ( )y f x g x=( )( )

f xyg x

= . Are the combined

functions even, odd or neither? d) Use graphing technology to confirm

your answers in part c).

4. Use Technology The gross domestic product (GDP) of a country is modelled by , where G is the GDP, in billions of dollars, and t is time, in years from now. The population of the country is growing according to

, where P is population, in millions, and t is time, in years, from now.

( )4( ) 24 1.025 tG t =

( )4( ) 3 1.04 tP t =

a) Graph G and P on the same set of axes, and describe their trends over the next 45 years.

b) Graph ( )( )

G tyP t

= on a new set of axes,

and describe its trend.

c) ( )( )

G tyP t

= gives the GDP per person

and provides a measure of the efficiency of workers. Calculate the GDP per person at the present time, and 5 years from now. Use your answer to part b) to describe what happens to worker efficiency as time goes on.

d) If the GDP per person falls below $4500, a recession is predicted. Will a recession occur in this country? If so, when?

5. Use Technology The horizontal distance

of a child’s swing from its resting position is modelled by ,

where

( ) ( )y A t p t=

( ) 6( ) 3 2t

A t −= gives the amplitude of the vibration, in metres,

2π( ) sin3

p t ⎛= ⎜⎝ ⎠

t ⎞⎟ governs the back-and-

forth part of the swing’s motion, and t is time, in seconds. a) From what you know by watching a real

swing, sketch a possible position time graph for the motion of a real swing over an extended period of time.


Name: _______________________________ Date: ________________________

b) Graph , over a 20-s time

period. Comment on any differences between this graph and your answer to part a).

( ) ( )y A t p t=

c) What seems to be the effect of multiplying a sine function by the amplitude function?

6. The algebraic tests used to decide

whether a function is even or odd are as follows. • A function f is even provided

( ) ( )f x f x− =

( ) (

. • A function f is odd provided

)f x f− = − x . a) Suppose f and g are both odd. Prove

that is even. ( ) ( )y f x g x=b) Suppose f is even and g is odd. Prove

that is odd. ( ) ( )y f x g x=c) Suppose f and g are both even. Prove

that is even. ( ) ( )y f x g x=d) Is the product of functions in any way

analogous to the multiplication of numbers when it comes to evenness and oddness? Explain.

…BLM 8–3. . (page 2)

7. a) Graph ( ) 63 2x

y −= and

( ) 62π3 2 sin3

x

y − ⎛= ⎜⎝ ⎠

x ⎞⎟ on the same set

of axes. b) Graph 5y x= ± + and

( )5 cosy x= + x on the same set of

axes.

c) Graph 5y x= ± + and 5cosxy

x+

=

on the same set of axes. d) Describe how the graphs for parts a),

b) and c) are similar.


Name: _______________________________ Date: ________________________

8.3 Composite Functions …BLM 8–4. .

1. Let ( ) 2 1f x x= + and 2( )g xx

= .

a) Determine a simplified algebraic model for each.

i) ( ( ))y f g x= ii) ( ( ))y g g x=b) Evaluate. i) ( (1))y f f= ii) ( (3))y g f=

2. The number of items, n(t), made per

week by a company is given by , where t is time, in

weeks. The profit of the company, in thousands of dollars, depends on the number of items produced, according to the formula

( ) 2000 40n t t= +

( )P n 3 1000n= − . a) Determine the equation for weekly

profit of the company. b) Use graphing technology to graph the

weekly profit for the next 2 years. c) Use algebraic techniques to determine

when the weekly profit first reaches $180 000.

d) Use a graphical method to check your answer to part c).

3. Let 2( )1

f xx

=+

and ( ) 3g x x= + .

a) Write the equations for and .

1( )y f x−=1( )y g x−=

b) Determine a simplified algebraic model for each.

i) 1 1( )y f g x− −= o ( )( )

))

ii) 1 1( )y g f x− −= o

iii) 1( ) (y f g x−= o− iv) 1( ) (y g f x= o

c) Which of the equations in part b) are the same?

4. Let 2( )f x x= and ( ) cosg x x= . a) Write the equation for . ( ( ))y f g x=b) Use graphing technology to graph

( ( ))y f g x= . c) Is ( ( ))y f g x= periodic? d) Write the equation for . ( ( ))y g f x=e) Use graphing technology to graph

( ( ))y g f x= . f) Is ( ( ))y g f x= periodic?

5. Define nf by the rule that

( )( )( )( ) ... ( )n

n times

f x f f f f f x=144424443

(

. For

example, )( )3( ) ( )f x f f f x= .

a) If ( )f x = x , calculate . 4 (65 536)f

b) For 1( )f xx

= , calculate each value.

i) 2 (5)f ii) 4 (5)f

200 iii) (5)f1001 iv) (5)f

c) Determine what happens to the value of ( )nf x as n for each function.

→ +∞

i) ( )f x x= ii) 2( ) , 0 1f x x x= < < iii) 2( ) , 1f x x x= =

2

iv) ( ) , 1f x x x= >


Name: _______________________________ Date: ________________________

8.4 Inequalities of Combined Functions …BLM 8–5. . (page 1) 1. Consider the functions 2( )f x x= and

. The functions are shown on the graph below.

( ) 2 3g x x= +

a) For what values of x is ( ) ( )f x g x> ? b) Sketch the graph of . ( ) ( )y f x g x= −c) For what interval is the function

f g− positive? d) Explain why your answers to parts a)

and c) are the same.

2. Let 31( )4

f x = x and ( )g x x= .

a) Graph f and g on the same set of axes. b) For what values of x is ( ) ( )f x g x< ?

c) Graph the function ( )( )

f xyg x

= on the

same axes as the original functions. d) Add the line y = 1 to your graph. e) Calculate the x-value(s) for the

point(s) of intersection of ( )( )

f xyg x

=

and y = 1. f) Is the answer for part e) related to the

answer for part b)? Explain.

3. A company makes solar panels. The company’s revenue function, in dollars, is ( ) 10R n n= , where n is the number of panels produced. The cost function is

( )30( ) 100 2C n =n

. R and C are shown on the graph below.

a) Estimate from the graph i) the break-even points ii) the number of panels that should be

produced for maximum profit b) Write the equation for the profit

function P. c) Graph P using graphing technology. d) Use your graph of P to estimate the

number of panels that give maximum profit.

e) How would your answers for break-even points and maximum profit change if

i) the number of dollars of revenue per panel is increased slightly?

ii) the cost function is changed to

( )35( ) 100 2n

C n = ? f) What does the number that was

changed in part e) ii) represent?


Name: _______________________________ Date: ________________________

4. In the early 19th century, Thomas Malthus

published An Essay on the Principle of Population. He argued that, since food supply increases linearly but population growth is exponential, people in the future are doomed to starvation. Consider a food supply model and a

population model , where t represents years from now. Starvation occurs when

( ) 40 1000F t t= +

(( ) 500 1.P t =

( ) ( )

)04 t

F t P t< . a) Calculate the rate of increase of the

food supply. What percent of the original supply is this?

b) Calculate the rate of population increase.

c) Use a graphical method to decide whether starvation occurs in this model, and, if so, when?

d) Is there any rate of increase in the food supply that will prevent starvation under this model? Try using graphing technology to animate the rate.

e) Is there any non-zero rate of increase in the population that will prevent starvation? Try animating the rate of growth.

…BLM 8–5. . (page 2)


Name: _______________________________ Date: ________________________

8.5 Making Connections: Modelling with …BLM 8–7. . Combined Functions (page 1) For question 1 and 2, refer to Example 1 on page 463. 1. There are lots of major triad chords. One

is the D-major triad, consisting of D, F# and A. a) Graph the combined function formed

by this triad. b) Compare the waveform to the one for

the C-major triad. 2. When two sounds occur in the same area

that have almost the same frequency, a special phenomenon called ‘beats’ occurs. a) Graph the combined function formed

when a sound with frequency of 10 Hz combines with a sound of frequency 9 Hz over the domain

1 1x− ≤ ≤ . b) Describe the pattern of the waveform

(called an interference pattern). c) How far apart, in seconds, are the

maximums of the pattern? d) Since the intensity of a sound depends

on its amplitude, what do beats sound like?

3. As time goes on, the amplitude of the

vibrations of a vibrating object, such as a pendulum, gradually decreases, even though the frequency remains the same. The position of the object from its rest position is given by the combined

function ( ) (( ) 2 sin 2πtH )x t A ft−= , where

• A is the original amplitude of the vibration, in metres • H is the time it takes for the amplitude to decrease to half its original value • f is the frequency of the vibration • t is time, in seconds

Suppose a pendulum has a frequency of 0.0125 Hz, an original amplitude of 3 m and takes 30 s for its amplitude to be cut in half. a) Use technology to graph . ( )y x t=

b) On the same set of axes, graph

( )2tHy A −= , using the same values

for A and H.

c) What does the graph of ( )2tHy A −=

appear to do to the vibrations? 4. The following graphs are formed by the

sum of ( )5sin 2πy = x and one other function. Develop an algebraic model for each graph. a)

b)


Name: _______________________________ Date: ________________________

c)

…BLM 8–7. . (page 2)


Name: _______________________________ Date: ________________________

Chapter 8 Review …BLM 8–8. . 8.1 Sums and Differences of Functions 1. Use the graph shown.

Use the superposition principal to draw a

graph of each. a) ( ) ( )y f x g x= +b) ( ) ( )y g x f x= −

2. If , ( ) 2 3f x x= + 2( )g x x x= + , and

, develop an algebraic model for each of the following.

2( ) 3h x x= −

a) ( ) ( )y f x g x= −b) ( ) ( ) ( )y f x g x h x= − +

8.2 Products and Quotients of Functions 3. Let , , and

. Work in radians. ( )u x x=

2( ) x=

( ) sinv x x=

w xa) Will the graph of have

symmetry? Explain why or why not. ( ) ( )y u x v x=

b) Will the graph of have symmetry? Explain why or why not

( ) ( )y u x w x=

c) Use graphing technology to confirm your predictions.

d) Is the domain of the same as the domains of and

? Explain.

( ) ( )y u x v x=(y u= )x

( )y v x=

e) Is the domain of ( )( )

u xyv x

= the same

as the domains of and ? Explain.

( )y u x=( )y v x=

8.3 Composite Functions 4. If ( ) 2 3f x x= + , 2( )g x x x= + , and

2( ) 3h x x= − , develop algebraic models for each. a) ( ( ))y f h x= b) ( )(y g f x= o )

)

)

c) 1( ( )y f f x−= 8.4 Inequalities of Combined Functions 5. Use Technology Jay has $4000 to invest.

He could invest in a money fund, where the value is given by the compound interest formula . He could buy a guitar, whose value is modelled by .

(( ) 4000 1.06 tA t =

2) 80 800 400t t= − +( 0V ta) Graph the two options. b) Which option is better over the first

10 years? c) After how long would the value of

both investments be the same? 8.5 Making Connections: Modelling with Combined Functions 6. Relative motion uses the superposition

principle. Consider the position of a person relative to the ground as she runs back and forth in a train while the train itself is also moving. a) Suppose the runner’s motion relative to

the train is ( )1( ) 5cos 0.2πx t t= − , where

1x is in metres and t is in seconds. If the train is moving at a constant speed of 1 m/s, its position is 2 ( )x t t= .

i) Predict what the motion of the runner would look like from the vantage point of a person standing on the ground beside the train as it went by. Sketch a graph of this motion.

ii) Graph 1 2( ) ( )y x t x t= + to check your answer to part a). Work in radians, and use technology.

b) Repeat part a) but with the train accelerating from rest, so that

22 ( ) 0.15x t t= .

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 8–8 Chapter 8 Review

Name: _______________________________ Date: ________________________

Chapter 8 Test …BLM 8–10. . 1. The graphs of two functions are shown.

a) Evaluate. i) ii) ( 3) ( 3)f g− + − ( ( 3))g f −b) Draw the graph of . ( ) ( )y f x g x= +c) For what values of x is each true?

i) ( ) ( )g x f x> ii) ( ) 1( )

g xf x

>

d) For what value of x is ( )( )

g xf x

not defined?

2. The equations for the graph in question 1

are and . ( ) 2f x x= + ( )2( ) 2 6g x x= + −a) Calculate.

i) ii) ( (2))f g ( )3gf

b) Develop a simplified algebraic model for each.

i) ( ) ( )y f x g x= + ii) ( ) ( )y f x g x= iii) ( )(y f g x= o )

3. a) The radius of a circle is related to its

circumference, ( )2πCr C = . Since the

area of a circle is given by , the area can be written in terms of the circumference as . Determine the simplified algebraic model for

.

2( ) πA r r=

( ( ))A r C

( ( ))A r C

b) Repeat this technique to determine an expression for the surface area of a cube in terms of its volume. Let the length of a side be b.

4. The season ticket sales for a football

team depend on the number of wins that the team has in the previous year according to the formula

2

( ) 10 000 110wN w ⎛= +⎜

⎝ ⎠⎞⎟ , where the

number of season tickets sold is N and the number of wins in the previous season is w. The number of wins that the team has in the previous year is currently

given by ( )21( ) 4 122

w t t= − − + , where t

is the year, with t = 0 in 2009. a) Calculate the number of season tickets

sold in 2009. b) Graph ( )y w t= . If the season has 14

games, describe what the model predicts concerning the winning success of the team.

c) What domain restrictions must be placed on w(t)? Explain.

d) The team’s stadium holds 50 000 fans. Does the model predict a need for expansion? Justify your answer.

e) The team plans to sign several older star free agents for next year. How might this affect ? ( )w t

f) The team has just drafted four rookies who are projected to be stars in 4 years. How might this affect w(t)?

5. For the football team in question 4, the

average season ticket price is $400. a) Determine an algebraic model

(un-simplified) for revenue ( )R t from season ticket sales.

b) Graph the model for . 0 9t≤ ≤c) The cost of running the team is

( )( ) 12 300 000 1 0.05C t t= + . Add this function to the graph.

d) When will the team be profitable?

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 8–10 Chapter 8 Test

Chapter 8 Practice Masters Answers …BLM 8–12. . (page 1)

Prerequisite Skills 1. a)

b) quadratic c)

d) 22T n=

2. a) even values 2 ,k k ∈b) odd values 2 1 ,k k− ∈

3.

{ }, 2x x∈ ≠ − , { }, 0y y∈ ≠

4. a) 13

yx

=−

, 3, 1, 0x x y≠ ≠ − ≠

b) { }, 3, 1x x x∈ ≠ ≠ − ,

1, 0,4

y y y⎧ ⎫∈ ≠ ≠ −⎨ ⎬⎩ ⎭

c)

d) asymptote , y = 0, hole at 3x =

11, 4

⎛ ⎞− −⎜ ⎟⎝ ⎠

5.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 8–12 Chapter 8 Practice Masters Answers

6. a) ( )1 5 2f x x− = ± − −

b) ( )1 1 24

f xx

− = +−

7. The inverse of 6 b) is a function, because the original function is 1 to 1.

8.1 Sums and Differences of Functions 1. a) h(x) = 5x − 2 b) 8

c) k(x) = x − 8 d) –6 2. a)

b) { }x∈ , { }, 7.5x x∈ ≤ c) { }x∈ , { }, 1.5y y∈ ≥

3. a)

b) { }, 0x x∈ ≠ , { }y∈ c) ( ) 2f x x= , as ( ) 2,x f x x→+∞ →

4. a) C(n) = 1500 + 200n b) R(n) = 250n c)

d) 30 e) P(n) = 50n − 1500


f) { },0 200n n∈ ≤ ≤

{,

}Range: 1500, 1450− − , 1400, ,8500− L g) $175 per unit sold

5. a)

b)

c) 2 sin4

y x π⎛ ⎞= +⎜ ⎟⎝ ⎠

d)

e) 2 sin4

y x π⎛ ⎞= −⎜ ⎟⎝ ⎠

6. a)

b) a square wave

c) a saw-tooth wave

8.2 Products and Quotients of Functions 1. a) for f(x): { }x∈ , { }y∈ , for g(x):

{ }, 2x x∈ ≠ ,{ }, 0y y∈ ≠ b)

c) for fg: { }, 2x x∈ ≠ ,{ }, 2y y∈ ≠ ;

for fg

: { }, 2x x∈ ≠ ,

{ }, 0,y y y 2∈ ≠ ≥ − d) for fg , vertical asymptote ,

horizontal asymptote ;

for

2x =2y =

fg

, hole at (2, 0)

2. a) for f(x): { }, 5x x∈ ≥ − ,

{ }, 0y y∈ ≥ ; for g(x): { }x∈ ,

{ }, 1 1y y∈ − ≤ ≤

b) ( ) ( ) ( )5 cosf x g x x x= + ,

( )( )

5cos

f x xg x x

+=



c)

d) for fg: { }, 5x R x∈ ≥ − , { }y R∈ ; for

fg

:

( ) π, 5, 2x x x∈ ≥ ≠ 1 ,2

k k⎧ ⎫− ∈⎨ ⎬⎩ ⎭

,

{ }, 0 or 1.30663y y y∈ ≥ ≤ − e) vertical asymptotes at

( ) π2 1 ,2

x k k= − ∈

3. a) “V”-shaped, even

b) parabolic, even

c) both even d) ( ) ( )y f x g x=

( )( )

f xy

g x=


4. a) both increase, but P increases faster after about 36 years

b) decreasing, approaching 0

c) $8000/person, $5983/person d) yes, after approximately 10 years

5. b)

c) makes the sine curve’s amplitude get

smaller 6. d) for functions, odd odd = even,

odd ×

× even = odd, and even × even = even. This is exactly the opposite of multiplication of numbers.

7. a)

b)

c)

d) The first functions form “boundaries”

for the combination functions.


8.3 Composite Functions

1. a) i) 4 1yx

= + ii) y = x

b) i) 7 ii) 27

2. a) ( ) 3 1000 40P t t= + b)


c) 65 weeks d)

3. a) ( )1 2 1f x

x− = − , ( )1 3g x x− = −

b) i) ( )( )1 1 2 13

f g xx

− − = −−

o

ii) ( )( )1 1 2 4g f xx

− − = −o

iii) ( ) ( )1 2 4f g xx

− = −o

iv) ( ) ( )1 2 13

g f xx

− = −−

o

c) ( )( ) ( ) ( )11 1g f x f g x−− − =o o

(,

)( ) ( ) ( )11 1f g x g f x−− − =o o

2

4. a) cosy x=b)

c) yes, period π d) ( )2cosy x=

e)

f) no

5. a) 2 b) i) 5 ii) 5 iii) 5 iv) 0.2 c) i) 0 ii) 0 iii) 1 iv) ∞

8.4 Inequalities of Combined Functions 1. a) 1x < − or 3x >

b)

c) x < −1 or x > 3 2. a)

b) ( ), 2 −∞ − or (0, 2) c)

e) –2, 2 f) yes, when y = 1, ( ) ( )f x g x= , which is where ( )f x

( )

changes from being less than g x to greater than ( )g x .

3. a) i) 14 and 99 panels ii) 63 panels

b) ( ) ( )3010 100 2n

P n n= − c)


d) 63 e) i) the first break-even point would be less, the second break-even point would be greater, maximum profit would increase ii) same effect as in i) f) the number of panels produced in order to double the original cost

4. a) 40 units per year, 4% b) 4% c) occurs in 43 years d) no e) no

8.5 Making Connections: Modelling With Combined Functions 1. a)

b) very similar

2. a)

b) amplitude regularly increases and decreases c) 1 s d) pattern of loud-soft-loud-soft…

3. a)


b)

c) it forms a boundary for the wave

4. a) ( ) 25sin 2πy x= + x b) ( )5sin 2π 2y x= + x c) ( )5sin 2π 2xy x= +

Chapter 8 Review 1. a)

b)

2. a) 2 3y x x= − + + b) y = x 3. a) Yes, since both original functions are

odd, the product function will be even. b) Yes, since one original function is odd

and the other even, the product will be odd.

d) Yes, when you multiply numbers there is no restriction.

e) No, when you divide numbers, you cannot divide by 0. Any domain elements for ( ) sinv x x= that yield y = 0 must be excluded from the domain

of ( )( )

u xy

v x= .

4. a) 22 3y x= − b) 24 14 1y x x 2= + + c) y = x

5. a)

b) money fund c) 14.6 years


6. a) ii)

b) ii)

Chapter 8 Test 1. a) i) –6 ii) −5

b)

c) i) ( ) ( ), 4 or 1,−∞ − +∞

(

ii) (−4, −2) or )1,+∞ d) –2

2. a) i) 12 ii) 195

b) i) ( ) ( ) 2 5f x g x x+ = +


x ii) ( ) ( ) 3 26 6f x g x x x x= + + − 4 iii) ( )( ) 2 4f g x x x= +o

3. a) ( )2

4πCA C = b)

23( ( )) 6A b V V=

4. a) 19 600 b)

Initially a poor team, winning only 4 of the 14 games in the first season, then becomes a good team after 4 years, then becomes poor again.

c) 0 9 starts in 2009, number of wins cannot be less than 0

t≤ <

d) No. ( )( )4 48 40N w = 0 e) It will make team better right away, so

would move vertex to the left.

f) It will make team better in 4 years, so would move vertex up.

5. a)

( )( )

221 4 12

24 000 000 110

tR t

⎛ ⎞− − +⎜ ⎟= +⎜ ⎟

⎜ ⎟⎝ ⎠

b)

c)

d) 2010 to 2015

chapter 8 prerequisite skillspjk.scripts.mit.edu/lab/mhf/chapter_8_practice.pdfblm 8–3 section 8.2...

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