chapter 3 prerequisite skillspjk.scripts.mit.edu/lab/mhf/mhf_chapter_3_all.pdf · chapter 3...

Name: _______________________________ Date: ________________________

Chapter 3 Prerequisite Skills …BLM 3–1. . Reciprocal Functions

1. Explain how the asymptotes of f(x) = 1x

relate to restrictions on the domain and range of f.

2. For each reciprocal function, write equations for the vertical and horizontal asymptotes. Use transformations to sketch the graph of each function relative to the graph of the base function

f(x) = 1x

.

a) 1( )2

f xx

=+

b) 1( )1

f xx

= −−

c) 1( )3 12

f xx

=−

Domain and Range 3. Write the domain and range for each

function. a) 2( ) 4f x x= −b) ( ) 3 4f x x= +c) ( ) 3 5f x x= − + −

d) 3( )2

f xx

= −+

Slope 4. Calculate the slope of the line that passes

through the points in each pair. Express your answer as an integer or a fraction in lowest terms. a) (–2, 5) and (1, 3) b) (4, –3) and (7, 3) c) (0, –3) and (–2, 0) d) (–2.3, 5) and (1.2, 2) e) (4.3, 2.7) and (2.6, –3.3)

5. Calculate the slope of the line that passes through the points in each pair. Express your answer as a decimal, rounded to two decimal places, when necessary. a) (4, –3) and (2, –4) b) (–3, –1) and (6, –3) c) (1.5, 2.6) and (3.2, –1.2) d) (1.63, –3.43) and (–4.15, 3.11)

Factoring Polynomials 6. Factor fully for x ∈ .

a) 2 3 4x x+ − b) 26 7x x 3− − c) 212 6x x+ − d) 3 24 7 14x x x 3− − − e) 38 12x + 5

0

Solving Quadratic Equations 7. Determine the roots of each quadratic

equation. a) 2 2 35x x+ − = b) 23 11 4x x 0− − = c) 26 11 4x x 0− + = d) 212 31 20 0x x+ + =

8. Determine the x-intercepts, if any exist.

Express your answers in exact form. a) 2 2 5y x x= + − b) 2 4 1y x x= − + c) 22 7y x x= + − d) 2 3 5y x x= − + −

Solving Inequalities 9. Solve each inequality. Show your

answers on a number line. a) 2 16x < b) 2 2 15 0x x− − ≥ c) 22 7 1x 0+ ≤ d) 2 23 3 2 6 3x x x x− − ≥ − + e) 2 23 6 2x x x x 3+ ≥ + − f) 22 33 16x x+ ≥

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

Name: _______________________________ Date: ________________________

3.1 Reciprocal of a Linear Function …BLM 3–2. . (page 1) 1. Copy and complete the table to describe

the behaviour of the function 1( )

4f x

x=

−.

As x→ f(x)→ 4+ 4− +∞ −∞

2. Write equations to represent the

horizontal and vertical asymptotes of the rational function. Then, write a possible equation for the function.

3. For the function 2( )4

f xx

=−

,

a) write equations to represent the vertical and horizontal asymptotes

b) determine the y-intercept

4. Determine a possible equation to

represent each function shown. a)

b)

5. Sketch each function and then describe

the intervals where the slope is increasing and the intervals where it is decreasing.

a) 1( )3

f xx

= −+

b) 2( )2 3

h xx

=−

6. Sketch a graph of each function. Label

the y-intercept. State the domain, the range, the equations of asymptotes, and the intervals over which the slope is increasing and decreasing.

a) 1( )5

f xx

= −−

b) 2( )5 2

h xx

=−

7. The pressure inside a cylinder is

inversely proportional to the volume of the gas inside it. When the volume of gas is 50 cm3, the pressure is 400 kPa. a) Write a function to represent the

pressure as a function of the volume. b) Sketch a graph of this function. c) Calculate the pressure for a volume of

75 cm3. d) As the volume increases, what

happens to the rate of change of pressure?

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 3–2 Section 3.1 Practice

Name: _______________________________ Date: ________________________

8. Investigate a variety of functions of the

form ( )2

bf xx

=+

, where . 0b >

a) What is the effect on the graph as the value of b is varied?

b) Use the results from your investigation to sketch a graph of each function.

i) 1( )2

f xx

=+

ii) 3( )2

f xx

=+

iii) 5( )2

f xx

=+

9. Analyse the key features (domain, range,

vertical asymptotes, and horizontal

asymptotes) of 1( )sin

f xx

= , and then

sketch the function.

…BLM 3–2. . (page 2)


Name: _______________________________ Date: ________________________

3.2 Reciprocal of a Quadratic Function …BLM 3–3. . (page 1) 1. Copy and complete the table to describe

the behaviour of the function

( )( )1( )

2 5f x

x x=

+ +.

As x→ f(x)→ 2+− 2−− 5+− 5−− +∞ −∞

2. Determine the equations for the vertical

asymptotes, if they exist, for each function. Then, state the domain.

a) 21( )7 6

f xx x

= −− +

b) 21( )4 6

f xx x

=+ +

3. Make a summary table with the headings

shown for each graph. Then, determine a possible equation for each graph.

Interval Sign

of f(x) Sign of Slope

Change in Slope

a)

b)

4. For each function, i) determine the equations for the

asymptotes, if they exist ii) give the domain iii) determine the x- and y-intercepts, if

they exist iv) sketch a graph of the function v) give a summary table of the slopes vi) give the range vii) approximate the slope of the graph at

the y-intercept

a) ( )( )

12 4

yx x

=− +

b) ( )2

14

yx

=+

c) 21

4y

x=

+

5. Sketch a graph of each function.

a) 21

4y

x= −

−

b) 215 4

yx x

=− +

c) 21

2 3y

x x=

− −

d) 21

4y

x= −

+

6. State the coordinates of the maximum or

minimum point for each of the graphs in question 5.


Name: _______________________________ Date: ________________________

7. The apparent brightness of a light source

is inversely proportional to the square of the distance from the light source. At a distance of 2.4 m, the brightness of a particular light source is 500 lux. a) Determine an equation relating the

brightness of the light source and the distance from the source.

b) Sketch a graph of the relationship. c) What is the brightness of the light

source at a distance of 12 m? d) Determine the range of distances for

which the brightness of the light source is less than 100 lux.

…BLM 3–3. . (page 2) 8. One method of graphing rational

functions that are reciprocals of polynomial functions is to sketch the polynomial function and then plot the reciprocals of the y-coordinates of key ordered pairs. Use this method to sketch

the graph of y = 1( )f x

for each function.

a) ( )( )( )

1( )2 2 4

f xx x x

=− + +

b) ( )( )2

1( )2 2

f xx x

=− +


Name: _______________________________ Date: ________________________

3.3 Rational Functions of the Form f(x) = ax + bcx + d …BLM 3–4. .

(page 1) 1. For each function,

i) determine the equations of the asymptotes

ii) state the domain and range iii) sketch the graph iv) summarize the increasing and

decreasing intervals

a) 2( )3

xf xx+

=+

b) 4( )2

xg xx

=−

c) 2 3( )4 3

xh xx

+=

−

2. a) State the equations of the vertical and

horizontal asymptotes of

( ) ax bf xcx d

+=

+.

b) State the domain and range of

( ) ax bf xcx d

+=

+.

3. Determine an equation in the form

( ) ax bf xcx d

+=

+ for the function shown in

each graph. a)

b)

4. Determine an equation of the form

( ) ax bf xcx d

+=

+ for a rational function

whose graph has the indicated features. a) vertical asymptote x = 3, horizontal

asymptote y = –2, and passing through the point (2, 1)

b) vertical asymptote x = 4, horizontal asymptote y = 3, and x-intercept of 3

5. The cost of an appliance whose purchase

price is $600 and annual hydro cost is $115 is given by the function

600 115( ) nC nn+

= , where C is the

annual cost, in dollars, and n is the number of years. a) Sketch a graph of C versus n. b) As n becomes very large, what

happens to C? c) Determine the number of years needed

to reduce the annual cost to below $200.


Name: _______________________________ Date: ________________________

6. Consider the rational function

1( ) 3 42

f xx

⎛ ⎞= +⎜ ⎟−⎝ ⎠.

a) Use transformations to compare the graph of the function to the graph of

1yx

= .

b) Rewrite the equation for f in the form

( ) ax bf xcx d

+=

+.

c) How are the values of a, b, c, and d related to the numbers in the original equation for f?

7. Repeat question 6 for the function

1( )f x k qx p

⎛ ⎞= +⎜ ⎟−⎝ ⎠

.

…BLM 3–4. . (page 2) 8. Use Technology The concentration of a

drug in the bloodstream is given by the

equation 25( )

0.01 3.3tC t

t=

+, where t is

the time, in minutes, and C is the concentration, in micrograms per millilitre. a) Graph the function using technology. b) Determine the maximum

concentration and when it will occur. c) Determine the effect of changing the

values of the coefficients in the equation.


Name: _______________________________ Date: ________________________

3.4 Solve Rational Equations and Inequalities …BLM 3–5. . 1. Solve algebraically. Check each solution.

a) 6 52 1x

=−

b) 6 5xx= −

c) 251

3 8x x=

− + 2

d) 5 11 3

x xx x+ +

=− −

2. Use Technology Solve each equation

using technology. Express your answers to two decimal places.

a) 2 52 3 1

x xx x−

=+ −

b) 2 5

3 2 1x x

x x+

=+ −

3. Solve each inequality without using

technology. Illustrate the solution on a number line.

a) 4 12 3 4x x

<− +

b) 2 3 6 53 3 1

x xx x+ −

≥− +

c) ( )( )( )( )

3 2 10

4 5x xx x− −

>+ −

d) 2

2

2 5 3 05 4

x xx x

+ −≤

+ +

4. Solve and check.

a) 3 4 05x x+ =

+

b) 32 5xx

= −

c) 2 31 1

1x x x+ =

− +

d) 3 25 01x x+ + =

−

5. Each inequality is of the form f(x) > g(x), where f(x) and g(x) are both rational functions. Graph f and g and use the graphs to solve each inequality.

a) 3 1

x xx x

>+ −

b) 2 3 1x xx x+ +

>

6. Jordan has a sister who is three years

older than he is, and a brother who is two years younger than he is. How old must Jordan be in order that the ratio of his sister’s age to his brother’s age is less than 2?

7. The amount of energy needed to increase the radius of orbit of a 500-kg satellite from its original orbit of radius 10 000 km can be modelled by the

function 10 10 0002 10 rEr

−⎛ ⎞= × ⎜ ⎟⎝ ⎠

, where

E is the energy, in Joules, and r is the new radius, in kilometres. a) Calculate the new radius of the

satellite if Joules of energy are added to it.

1010

b) How much energy must be given to the satellite in order for it to escape Earth’s gravity completely (make its orbit’s radius infinitely large)?

8. Shade the area of the Cartesian plane

where 2 2101

4x y

x+ ≤ ≤

+.

9. Solve for A and B:

2

13 12 3 2 2 1 2

x A Bx x x x

+= +

+ − − +.


Name: _______________________________ Date: ________________________

3.5 Making Connections With Rational Functions …BLM 3–7. . and Equations (page 1) 1. In order to create a saline solution, salt

water with a concentration of 40 g/L is added at a rate of 500 L/min to a tank of water that initially contained 8000 L of pure water. The resulting concentration of the solution in the tank can be

modelled by the function 40( )160

tC tt

=+

,

where C is the concentration, in grams per litre, and t is the time, in minutes. a) In how many minutes the saline

concentration be 20 g/L? b) Is there an upper limit to the

concentration in the tank? Explain. c) What restrictions must be placed on

the domain of C if the tank has a maximum capacity of 120 000 L?

2. A company finds that its sales since the

company started in 2000 can be modelled

by the function 2

2

20 800 300( )8 10 100t tS tt t+ +

=+ +

,

where S is the total sales, in millions of dollars, and t is the number of years since 2000. a) What were the sales in 2000? b) After many years, what does the

model predict sales will be? c) Calculate the years when the sales are

$9 million, algebraically. d) Use Technology Use technology to

graph of the model. During what year were sales highest?

e) If you were working in the human resources department for the company, would you recommend that the company hire more people based on this model? Explain your reasoning.

3. The weight (gravitational force) on a 100-kg object as a function of its height above mean sea level on Earth can be modelled by the formula

( )16

26

4 10( )6.4 10

W hh

×=

× +, where W is the

weight, in Newtons (1 kg weighs about 10 N) and h is the height above mean sea level, in metres. a) How much does the object weigh at

sea level? b) If you were to take the object to the

top of Mt. Everest (height 9000 m), what would its weight be?

c) How high would the object have to be to weigh 800 N? Round your answer to the nearest kilometre.

4. An integer n is squared, and the result

doubled. Three is added to the same integer and the result squared. The ratio of the first answer to the second is then formed. a) Write a function R(n) that gives the

ratio of the two answers. b) Sketch the graph of R. c) A student claims that the value of R

will always be less than 2. Is she correct? Explain.

d) Solve algebraically to determine the values of n for which R(n) ≤ 0.5. Illustrate your answer on a number line.

e) For which value(s) of n is R(n) > 8?


Name: _______________________________ Date: ________________________

5. A rectangular prism with a square base

has a volume of 25 cm3. The surface area of the prism is given by the formula

32 10( ) bS bb+

=0 , where S is the surface

area, in square centimetres, and b is the length of each side of the base, in centimetres. a) What is the restriction on the length of

the base? b) Use Technology Use technology to

graph the function S over the domain [0, 10].

c) Use Technology Use technology to calculate the length of the base that would give the smallest surface area.

d) This function has no asymptote, but does approach a curve that is a parabola. Determine the equation of that parabola.

…BLM 3–7. . (page 2)


Name: _______________________________ Date: ________________________

Chapter 3 Review …BLM 3–8. . (page 1) 3.1 Reciprocal of a Linear Function 1. Determine the equations for the vertical

and horizontal asymptotes of each function.

a) 1( )2 5

f xx

=−

b) 3( )5

g xx

= −+

2. Determine an equation to represent the

graph of the function.

3. Sketch a graph of each function.

a) 4( )2

f xx

=−

b) 1( )1

g xx

= −+

4. For each function, state

i) the domain and range ii) the x- and y-intercepts

a) 5( )4 7

f xx

= −−

b) 1( )5

g xx

=−

3.2 Reciprocal of a Quadratic Function 5. For each function,

i) determine the equations of the asymptotes

ii) determine the x- and y-intercepts iii) sketch the graph iv) state the domain and range v) list the intervals over which the

function is increasing vi) list the intervals over which the slope

of the graph is increasing

a) 1( )( 2)( 1

f xx x

=)+ −

b) 28( )

4g x

x= −

−

c) 24( )

4g x

x=

+

6. A function that is the reciprocal of a

quadratic function has vertical asymptotes x = –2 and x = 6. It has a horizontal asymptote y = 0 and the function is positive over the interval (–2, 6). Write an equation for this function.

3.3 Rational Functions of the Form

f(x) = ax + bcx + d

7. Summarize the key features of each function. Then, sketch a graph of the function.

a) 3( )2

xf xx−

=+

b) 4 3( )2 1

xg xx−

=+

8. A function of the form ( ) ax bf xcx d

+=

+

has the following features: • x-intercept –1

• y-intercept 32

• vertical asymptote x = –2 • horizontal asymptote y = 3 Determine an equation for this function.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 3–8 Chapter 3 Review

Name: _______________________________ Date: ________________________

9. Determine an equation of the function

whose graph is shown.

3.4 Solve Rational Equations and Inequalities 10. Solve algebraically.

a) 263

2 4x x=

− −

b) 2 3 2 75 3

x xx x− +

≥+ −

11. Use Technology Solve each equation

using technology. Round your answers to two decimal places, where necessary.

a) 2 25 2 1

3 3 1x x xx x− −

=+ −

+

b) 3

2

6 8 02

x xx x− +

<− −

…BLM 3–8. . (page 2) 3.5 Making Connections With Rational Functions and Equations 12. The population of a town can be

modelled by the function 4 3( ) 202 5

tP tt+⎛= ⎜

⎞⎟+⎝ ⎠

, where P is the

population, in thousands, and t is the time, in years, after the year 2000 (t > 0). a) What is the population in the year

2000? b) In what year will the population be

30 000? c) Town planners claim that they need

not plan for a population above 40 000. Does the model support this conclusion? Explain.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 3–8 Chapter 3 Review

Name: _______________________________ Date: ________________________

Chapter 3 Test …BLM 3–10. . 1. Sketch a graph of each function.

a) 1( )4

f xx

= −−

b) 2 3( )1

xg xx−

=+

c) 230( )7 10

h xx x

=− +

2. Write a possible equation for the function

in each graph. a)

b)

c)

3. For the function whose graph is shown in question 2, part c), state the following: a) the domain and range b) the intervals over which the function is

increasing c) the intervals over which the slope of

the graph is decreasing 4. Solve. Check your answer.

2 23 1 1xx x−

=− +

5. Solve Illustrate your answer on a number

line.

( )( )3 0

2 4x

x x+

>− +

6. The distance of an image from a lens in a

camera can be modelled by the function 5

5dD

d=

−, where D is the distance from

the lens to the image and d is the distance from the subject being photographed to the lens. Both D and d are measured in centimetres. In order for a photograph to be in perfect focus, the distance from the lens to the sensor must be the same as the distance of the lens to the image. The distance of the lens from the sensor can be adjusted to allow this to happen. a) Sketch a graph of the function. b) Suppose the maximum distance from

the lens to the sensor is 9 cm. How far away from the lens is the subject in this case?

c) How far should the lens be from the sensor in order to have a distant subject in focus?

d) Use Technology Use technology to determine the values of d for which the distance of the lens to the sensor can be less than 6.5 cm.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 3–10 Chapter 3 Test

Chapter 3 Practice Masters Answers …BLM 3–12. . (page 1)

Prerequisite Skills 1. Vertical asymptotes give restrictions on

the domain. Horizontal asymptotes give restrictions on the range.

2. a) 2, 0x y= − =

b) 1, 0x y= =

c) 4, 0x y= =

3. a) { }x∈ , { }, 4y y∈ ≥ −

b) { }x∈ , { }y R∈ c) { }, 3x x∈ ≥ − , { }, 5y y∈ ≤ − d) { }, 2x x∈ ≠ − , { }, 0y y∈ ≠

4. a) 23

− b) 2 c) 32

−

d) 67

− e) 6017

5. a) 0.5 b) –0.22 c) –2.24 d) –1.13

6. a) ( )( )4 1x x+ − b) ( )( )3 1 2 3x x+ − c) ( )( )4 3 3 2x x+ − d) ( )( )( )1 3 4x x x+ − +1 e) ( )( )22 5 4 10 25x x x+ − +

7. a) –7, 5 b) 1 ,43

−

c) 1 4,2 3

d) 5 4,4 3

− −

8. a) 1 6− ± b) 2 3±

c) 1 574

− ± d) none

9. a) 4 4x− < < b) or 3x ≤ − 5x ≥

c) 3 32 2

x− ≤ ≤ d) or 6x ≤ − 1x ≥

e) 5 132

x − −≤ or 5 132

x − +≥

f) x∈ 3.1 Reciprocal of a Linear Function 1.

As x→ f(x)→ 4+ +∞ 4− −∞ +∞ 0 −∞ 0

2. y = 0, x = –1, 11

y =x +

3. a) 4, 0x y= = b) 12

4. a) 12

yx

=−

b) 12 3

yx

=+

5. a) The slope is positive and increasing for x < –3. The slope is positive and decreasing for x > –3.

b) The slope is negative and decreasing

for x < 32

. The slope is negative and

increasing for x > 32

.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 3–12 Chapter 3 Practice Master Answers


6. a)

y-intercept 1

5, { }, 5x x∈ ≠ ,

{ },y y∈ ≠5, 0x y= =

0 , asymptotes: , slope positive and

increasing for x < 5, slope positive and decreasing for x > 5

b)

y-intercept 25

, 5,2

x x⎧ ⎫∈ ≠⎨ ⎬⎩ ⎭

,

{ }, 0y y∈ ≠ , asymptotes: 5 , 02

x y= = , slope positive and

increasing for x < 52

, slope positive

and decreasing for x > 52

7. a) ( ) 20 000P VV

=

b)

c) 800

3 kPa

d) As the volume increases, the rate of change of the pressure decreases.

8. a) 0 < b < 1, vertical compression by factor of b; b > 1, vertical stretch by factor of b

b) i)

ii)

iii)

9. { }, 180 ,x x k k∈ ≠ ∈o ,

{ }, 1, 1y y y∈ ≤ − ≥180 ,x k

, vertical asymptotes: k= ∈o

3.2 Reciprocal of a Quadratic Function 1.

As x→ f(x)→ 2+− +∞ 2−− −∞ 5+− −∞

5−− +∞ +∞ 0 −∞ 0

2. a) x = 1, x = 6; { }, 6, 1x x x≠ ≠ ∈b) none; { }x∈

3. a) Interval x < 2 x > 2 Sign of f(x) + + Sign of Slope + – Change in Slope + +

( )21

2y

x=

−



b) Interval x < 0 0 < x < 2 x = 2 2 < x < 4 x > 4 Sign of f(x) + – – – + Sign of Slope + + 0 – –

Change in Slope + – – +

( )5

4y

x x=

−

4. a) i) x = 2, x = –4, y = 0 ii) { }, 2, 4x x x∈ ≠ ≠ −

iii) no x-intercept, y-intercept 18

−

iv)

v)

Interval x < –4 –4 < x < –1 x = –1 –1 < x < 2 x > 2Sign of f(x) + – – – +

Sign of Slope + + 0 – –

Change in Slope + – – +

vi) 1, 0,9

y y y⎧ ⎫∈ > ≤ −⎨ ⎬⎩ ⎭

vii) –0.03 b) i) x = –4, y = 0 ii) { }, 4x x∈ ≠ −


iv)

v)

Interval x < –4 x > –4 Sign of f(x) + + Sign of Slope + – Change in Slope + +

vi) { }, 0y y∈ > vii) –0.03 c) i) y = 0 ii) { }x∈


iv)

v)

Interval x < 0 x = 0 x > 0 Sign of f(x) + + + Sign of Slope + 0 – Change in Slope + +

vi) 1,04

y y⎧ ⎫∈ < ≤⎨ ⎬⎩ ⎭

vii) 0 5. a)

b)

c)

d)

6. a) 10,4

⎛ ⎞⎜ ⎟⎝ ⎠

b) 5 4,2 9

⎛ ⎞−⎜ ⎟⎝ ⎠

c) 1 8,4 25

⎛ ⎞−⎜ ⎟⎝ ⎠

d) 10,4

⎛ ⎞−⎜ ⎟⎝ ⎠



7. a) 22880Bd

=

b)

c) 20 lux d) 28.8d > m

8. a)

b)

3.3 Rational Functions of the Form

f(x) = ax + bcx + d

1. a) i) x = –3, y = 1 ii) { }, 3x x∈ ≠ − , { }, 1y y∈ ≠ iii)

iv)

Interval x < –3 –3 < x < –2 x > –2 Sign of f(x) + – + Sign of Slope + + + Change in Slope + – –

b) i) x = 2, y = –4 ii) { }, 2x x∈ ≠ , { }, 4y y∈ ≠ − iii)

iv) Interval x < 0 0 < x < 2 x > 2 Sign of f(x) – + – Sign of Slope + + + Change in Slope + + –

c) i) 43

x = , 23

y = −

ii) 4,3

x x⎧ ⎫∈ ≠⎨ ⎬⎩ ⎭

, 2,3

y y⎧ ⎫∈ ≠ −⎨ ⎬⎩ ⎭

iii)

iv)

Interval x < –32 –

32 < x <

43 x >

43

Sign of f(x) – + – Sign of Slope + + + Change in Slope + + –

2. a) dxc

= − , ayc

=

b) , dx xc

⎧ ⎫∈ ≠ −⎨ ⎬⎩ ⎭

, , ay yc

⎧ ⎫∈ ≠⎨ ⎬⎩ ⎭

3. a) 2 12

xyx+

=−

b) 6 32 1

xyx−

=+

4. a) 2 33

xyx

− +=

− b) 3 9

4xyx−

=−

5. a)

b) as , n → +∞ 115C →c) approximately after 7 years

6. a) vertically stretched by a factor of 3, translated 2 units right and 4 units up

b) 4 5( )2

xf xx−

=−

c) c and d determine the translation of 2 units to the right; a and c determine the translation of 4 units up.



7. a) If k > 0, vertically stretched by a factor of k; if k < 0, flipped about x = p and vertically stretched by a factor of |k|; if p > 0, translated p units to the right; if p < 0, translated p units to the left; if q > 0, translated q units up; if q < 0, translated q units down

b) ( )( ) qx k pqf xx p+ −

=−

c) a = q, b = k – pq, c = 1, d = –p 8. a)

b) The maximum concentration is about

13.8 mg/mL, which occurs at 18.2 min.

c) Increasing the “5” vertically stretches the graph. Decreasing the “5” vertically compresses the graph. Changing the sign of the “5” flips the graph about the x-axis. Increasing the “0.01” vertically compresses the graph and shifts the maximum to the left. Decreasing the “0.01” vertically stretches the graph and shifts the maximum to the right.

3.4 Solve Rational Equations and Inequalities 1. a) x = 1.1 b) x = 6 or x = –1

c) x = 3 or x = 13

− d) x = 7

2. a) x 3.52 or x 0.28 b) x –0.61

3. a) x < 192

− or –4 < x < 32

b) 1 63 1

x− < ≤7

4

or 3x >

c) or x < − 1 32

x< < or 5x >

d) or 4 x− < ≤ −3 112

x− < ≤

4. a) x = 207

−

b) x = 1 or x = 32

or 3

c) x = 15

d) x = 0.4±

5. a) x < –3 or 0 < x < 1

b) x < –2 or x > 0 6. more than 7 years old 7. a) 20 000 km b) joules 102 10×8.

9. A = 3, B = 5 3.5 Making Connections With Rational Functions and Equations 1. a) 160 min

b) Yes. The upper limit is 40 g/L, because there is a horizontal asymptote.

c) 0 22t 4≤ ≤ 2. a) $3 million b) $2.5 million

c) approximately 2001 and 2013 d) 2003 e) Do not hire more people after 2003

because sales decline after that. 3. a) approximately 977 N

b) approximately 974 N c) 671 km

4. a) ( )( )

2

22

3nR n

n=

+

b)

c) This is true only for positive integers,

since as . , 2n R −→ +∞ →d) 1 3n− ≤ ≤

6 3n− < < − or 3 2n− < < −e)



5.

a) 0b > b)

c) approximately 2.9 cm

hapter 3 Review b) x = –5, y = 0

d) 22y x= C1. a) x = 2.5, y = 0

2. 3y = − 2x +

) 3. a

b)

4. a) i) 7,4

x x⎧ ⎫∈ ≠⎨ ⎬⎩ ⎭

, { }, 0y y∈ ≠

ii) no x-intercept, y-intercept 57

b) i) { }, 5x x∈ ≠ , { }, 0y y∈ ≠

x-intercept, y pt ii) no -interce 15

−

5.

pt

a) i) x = –2, x = 1, y = 0

ii) no x-intercept, y-interce 12

−

iii)

iv) { }, 2,x x x∈ ≠ − ≠ , 1

4, 0,9

y y y⎧ ⎫∈ > ≤ −⎨ ⎬⎩ ⎭

v) x 12

− < –2 or –2 < x <

vi) x < –2 or x > 1 b) i) x = –2, x = 2, y = 0 ii) no x-intercept, y-intercept 2 iii)

iv) { }, 2,x x x 2∈ ≠ − ≠ , { }, 0, 2y y y∈ < ≥ v) 0 < x < 2 or x > 2 vi) –2 < x < 2 c) i) 0y =

o x-intercept, y-intercept 1 iii) ii) n

iv) { }x∈ , { }| 0 1y y∈ < ≤ v) x < 0 vi)

6.)

( )(

1( )2 6

f xx x

= −+ −

7. = –2, 1, a) x y = { }, 2x x∈ ≠ − , { }, 1y y∈ ≠ , x-intercept 3,

32

−y-intercept ; for < –2, f(x) is

ncreasing and the slope is positive and increasing; for –2 < x < 3, f(x

ing

x

positive and i

) is negative andincreasing and the slope is positiveand decreas ; for x > 3, f(x) is positive and increasing and the slope is positive and decreasing



c) 12

− , b) x = y = 32

− , 1,2

x x⎧ ⎫∈ ≠ −⎨ ⎬⎩ ⎭

,

3,2

y y⎧ ⎫∈ ≠ −⎨ ⎬⎩ ⎭

, x-intercept 43

,

terc fory-in ept 4; x < 12

− , f(x) is

negative and decreasing and the slope is negative and decreasing;

12

− < x < 43

decreasing and the slope is negati

and increasing; for x >

, f(x) is positive and

ve 43

, f(x) is

negative and ope is n a

decreasing and the slegative nd increasing

8. 3 32

xyx+

=+

9. 23

xyx−

=+

10. a) 32

x = − or x = 2

b) x < –5 or –1 ≤ x < 3 a) x –1.81 or x 0.14 or x 7.66 b) or

12. a) 12 000 b) 2004 es; as

11..9

2 5x < − 1 2x− < <

c) Y ,→ +∞ → 40t P − .

Chapter 3 Test 1. a)

b)

b) 3 2

2xyx−

=+

2. a) 26

1y

x=

+

c))

( )(

94 2

yx x

=+ −

a) { }, 4, 2x x x∈ ≠ − ≠ , { }, 0,y y y 1∈ > ≤ −

3.

b) x < –4 or –4 < x < –1 c) –4 < x < 2

4. x = 0 or x = 7 5. 4 3x− < < − or 2x > 6. a)

b) 11.25 cm c) 5 cm d) greater than 21.7 cm


Chapter 3 Prerequisite Skills3.1 Reciprocal of a Linear Function3.2 Reciprocal of a Quadratic Function3.3 Rational Functions of the Form f(x) = (ax + b)/(cx + d)3.4 Solve Rational Equations and Inequalities3.5 Making Connections With Rational Functions and EquationsChapter 3 ReviewChapter 3 TestChapter 3 Practice Masters Answers

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