Answers! Lesson 5.1 Check Your...

Answers!

Lesson 5.1 Check Your Understanding

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1

Master 5.8

5.1 Relations pg 256-261 outcome R1 1. Animals can be associated with the classes they are in.

Animal Class

ant Insecta

eagle Aves

snake Reptilia

turtle Reptilia

whale Mammalia

a) Describe this relation in words.

b) Represent this relation:

i) as a set of ordered pairs

ii) as an arrow diagram

Solution

a) The relation shows the association “belongs to the class” from a set of animals

to a set of classes. For example, an eagle belongs to the class Aves.

b) i) The animals are the first elements in the ordered pairs.

The classes are the second elements in the ordered pairs.

The ordered pairs are: {(ant, Insecta), (eagle, Aves), (snake, Reptilia),

(turtle, Reptilia), (whale, Mammalia)}

ii) The animals are written in the first set of the AARRRROOWW diagram.

The classes are written in the second set; each class is written only once.




2

Master 5.8

average time in hours to drive to Vancouver

2. Different towns in British Columbia can be associated with the average time,

in hours, that it takes to drive to Vancouver.

Consider the relation represented by this graph. Represent the relation:

a) as a table

b) as an arrow diagram

Solution

a) The association is: “average time in hours to drive to Vancouver”

In the table, write the towns in the first column and the average times in hours

in the second column.

Town Average

Time (h)

Horseshoe Bay 0.75

Lillooet 4.5

Pemberton 2.75

Squamish 1.5

Whistler 2.5

b) In the arrow diagram, write the towns in the first set and the average times in hours

in the second set.




3

Master 5.8

3. In the diagram below:

a) Describe the relation in words.

b) List 2 ordered pairs that belong to the relation.

Solution

a) The relation shows the association “has this number of letters” from a set of English words

to a set of natural numbers.

b) Two ordered pairs that belong to the relation are:

(animal, 6), (umbrella, 8)

Homework Pg 262

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5-7,9

14




4

Master 5.9

5.2 Functions Page 264-270 outcome R2

1. For each relation below:

• Determine whether the relation is a function. Justify your answer.

• Identify the domain and range of each relation that is a function.

a) A relation that associates a number with a prime factor of the number:

{(4, 2), (6, 2), (6, 3), (8, 2), (9, 3)}

b)

Solution

a) Check to see if any ordered pairs have the same first element:

{(4, 2), (6, 2), (6, 3), (8, 2), (9, 3)}

Two ordered pairs have the same first element.

So, the set of ordered pairs does not represent a function.

b) Check to see if any element in the first set

associates with more than one element

in the second set.

Each element in the first set associates with

exactly one element in the second set; that is,

there is exactly one arrow from each month in the first set.

So, the relation is a function.

The domain is the set of first elements:

{January, February, March, April}

The range is the set of associated second elements:

{28, 30, 31}




5

Master 5.9

2. The table shows the costs of student bus tickets, C dollars, for different numbers

of tickets, n.

Number of Tickets,

n

Cost, C

($)

1 1.75

2 3.50

3 5.25

4 7.00

5 8.75

a) Why is this relation also a function?

b) Identify the independent variable and the dependent variable. Justify your choices.

c) Write the domain and range.

Solution

a) For each number in the first column, there is only one number in the second column. So,

the relation is a function.

b) Since the cost, C dollars, depends on the number of tickets, n, C is the dependent variable

and n is the independent variable.

c) The first column of the table is representative of the domain.

The domain is: {1, 2, 3, 4, 5, …}; that is, all natural numbers.

The second column of the table is representative of the range.

The range is: {1.75, 3.50, 5.25, 7.00, 8.75, …}; that is, the product of each natural number

and 1.75.




6

Master 5.9

3. The equation C = 25n + 1000 represents the cost, C dollars, for a feast following an Arctic

sports competition, where n is the number of people attending.

a) Describe the function. Write the equation in function notation.

b) Determine the value of C(100). What does this number represent?

c) Determine the value of n when C(n) = 5000. What does this number represent?

Solution

a) The cost of the feast is a function of the number of people attending.

In function notation: C(n) = 25n + 1000. C is a function of n

b) To determine C(100), use:

C(n) = 25n + 1000 Substitute: n = 100

C(100) = 25(100) + 1000

C(100) = 2500 + 1000

C(100) = 3500 if 100 attend it costs $3500

C(100) is the value of C when n = 100.

This means that when the number of people attending is 100,

the cost of the feast is $3500.

c) To determine the value of n when C(n) = 5000, use:

C(n) = 25n + 1000 Substitute: C(n) = 5000

5000 = 25n + 1000 Solve for n.

4000 = 25n Divide each side by 25.

n = 160

C(160) = 5000 means that when n = 160, C = 5000; that is, when the number of people

attending is 160, the cost of the feast is $5000.

Homework pg 270

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#8-10, 13-16, 19

#21




7

Master 5.10

5.3 Interpreting & Sketching Graphs page 276-281 outcome R1

April 4 2012

1. Each point on this graph represents a person.

Explain your answer to each question below.

a) Which person is the oldest? What is her or his age?

b) Which person is the youngest? What is her or his age?

c) Which two people have the same height? What is this height?

d) Which two people have the same age? What is this age?

e) Which of person B or C is taller for her or his age?

Solution

a) Person G is the oldest because he or she is represented by the point on the graph farthest to

the right and the horizontal axis represents age. Person G is 18 years old.

b) Person A is the youngest because he or she is represented by the point on the graph farthest

to the left and the horizontal axis represents age. Person A is 0 years old, a newborn baby.

c) Persons B and C have the same height because the points that represent them lie on the

same horizontal line and it passes through 100 on the Height axis. The height

is 100 cm.

d) Persons D and E have the same age because the points that represent them lie on the same

vertical line and it passes through 10 on the Age axis. The age is 10 years.

e) Person B is taller for her or his age because he or she is the same height as Person C, but 1

year younger




8

Master 5.10

2. This graph represents a day trip from Athabasca to Kikino in Alberta,

a distance of approximately 140 km. Describe the journey for each segment of the graph.

Solution

Segment Graph Journey

OA The graph goes up to the right, so as

time increases, the distance from

Athabasca increases.

The car leaves Athabasca and takes 2 h to

travel 140 km to Kikino. 140/2=70km/h

AB The graph is horizontal, so as time

increases, the distance stays the same.

The car stops for 1 h.

BC The graph goes down to the right, so

as time increases, the distance

decreases.

The car starts the return trip. The car takes

approximately 45 min to travel 50 km

toward Athabasca.

CD The graph is horizontal, so as time

increases, the distance stays the same.

The car stops for approximately 45 min.

DE The graph goes down to the right, so

as time increases, the distance

decreases.

The car takes 1 h to travel approximately

90 km to Athabasca.




10

Master 5.11

3. At the beginning of a race, Alicia took 2 s to reach a speed of 8 m/s.

She ran at approximately 8 m/s for 12 s, then slowed down to a stop in 2 s.

Sketch a graph of speed as a function of time. Label each section of your graph, and explain

what it represents.

Solution

Draw and label axes on a grid. The horizontal axis represents time in seconds.

The vertical axis represents speed in metres per second.

Segment Journey

OA Alicia’s speed increases from 0 to 8 m/s, so the segment

goes up to the right for the first 2 s.

AB Alicia runs at approximately 8 m/s for 12 s. Her speed does

not change, so the segment is horizontal.

BC Alicia slows down to 0 km/h in 2 s, so her speed decreases

and the segment goes down to the right.

Homework page 281

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#4-9,13,14

#16




11

Master 5.11

5.5 Graphs of Relations and functions pg 287-293 outcome R2

1. Which of these graphs represents a function? Justify your answer.

a) b)

Solution

Use the vertical line test for each graph.

a) This graph does represent a function. Any vertical line drawn on the graph passes through

0 points or 1 point.

b) This graph does not represent a function because two points lie on the same

vertical line.




11

Master 5.11

2. Determine the domain and range of the graph of each function.

a) b)

Solution

a) The dot at the right end of the graph indicates that

the graph stops at that point. There is no dot at the left

end of the graph, so the graph continues to the left.

The domain is the set of x-values of the function.

Visualize the shadow of the graph on the x-axis.

The domain is the set of all real numbers less than

or equal to 5; that is, x ≤ 5.

The range is the set of y-values of the function.

Visualize the shadow of the graph on the y-axis.

The range is the set of all real numbers less than

or equal to 2; that is, y ≤ 2.

b) The dot at each end of the graph indicates that

the graph stops at that point.

The domain is the set of x-values of the function.

Visualize the shadow of the graph on the x-axis.

The domain is the set of real numbers between –3 and 5,

including these numbers; that is,

–3 ≤ x ≤ 5.

The range is the set of y-values of the function.

Visualize the shadow of the graph on the y-axis.

The range is the set of real numbers between 3 and 7,

including these numbers; that is,

3 ≤ y ≤ 7.




14

Master 5.12

3. This graph shows the approximate height of the tide, h metres,

as a function of time, t, at Port Clements, Haida Gwaii on June 17, 2009.

a) Identify the dependent variable and the independent variable. Justify your choices.

b) Why are the points on the graph connected? Explain.

c) Determine the domain and range of the graph.

Solution

a) The approximate height of the tide is a function of time. Since the height of the tide,

h metres, depends on the time of day, the dependent variable is h and the independent

variable is t.

b) The points on the graph are connected because the height of the tide is not restricted to a

whole number. This means that the values between the points are valid; for example,

between 10:00 and 11:00 the height of the tide increases from 1.2 m to 1.4 m. The height

increases gradually, and has values between 1.2 m and 1.4 m.

c) The domain is the set of times; that is, 09:00 ≤ h ≤ 16:00

The range is the set of heights of the tide; that is, 0.9 ≤ h ≤ 1.5

4. Here is a graph of the function g(x) = 4x – 3.

a) Determine the range value when the domain value is 3.

b) Determine the domain value when the range value is –7.

Solution




15

Master 5.12

The domain value is a value of x. The range value is a value of g(x).

a) To determine the value of g(x) when x = 3:

Begin at x = 3 on the x-axis.

Draw a vertical line to the graph, then a horizontal line to the y-axis.

The line appears to intersect the y-axis at 9.

So, g(3) = 9

When the domain value is 3, the range value is 9.

b) To determine the value of x when g(x) = –7:

Since y = g(x), begin at y = –7 on the y-axis.

Draw a horizontal line to the graph, then a vertical line to the x-axis.

The line appears to intersect the x-axis at –1.

So, when g(x) = –7, x = –1

When the range value is –7, the domain value is –1.

5.5 Homework pg 294

#4-7

#8,9,11,12,15,17

#22, 24




16

Master 5.12

5.6 Properties of Linear Relations pg 300-307 outcome R4

1. Which table of values represents a linear relation? Justify your answer.

a) The relation between the number of bacteria in a culture, n, and time, t minutes.

t n

0 1

20 2

40 4

60 8

80 16

100 32

b) The relation between the amount of goods and services tax charged, T dollars, and the

amount of the purchase, A dollars

A T

60 3

120 6

180 9

240 12

300 15

Solution

The terms in the first column are in numerical order.

So, calculate the change in each variable.

a)

t Change in t n Change in n

0 1

20 20 – 0 = 20 2 2 – 1 = 1

40 40 – 20 = 20 4 4 – 2 = 2

60 60 – 40 = 20 8 8 – 4 = 4

80 80 – 60 = 20 16 16 – 8 = 8

100 100 – 80 = 20 32 32 – 16 = 16

The changes in t are constant, but the changes in n are not constant.

So, the table of values does not represent a linear relation.

b)

A Change in A T Change in T

60 3

120 120 – 60 = 60 6 6 – 3 = 3

180 180 – 120 = 60 9 9 – 6 = 3

240 240 – 180 = 60 12 12 – 9 = 3

300 300 – 240 = 60 15 15 – 12 = 3

Since the changes in both variables are constant, the table of values represents

a linear relation.




15

Master 5.12

2. a) Graph each equation.

i) x = –2 ii) y = x + 25 iii) y = 25 iv) y = x2 + 25

b) Which equations in part a represent linear relations? How do you know?

Solutions

a) i) x = –2

x y

–2 –1

–2 0

–2 1

ii) y = x + 25

x y

–10 15

–5 20

0 25

5 30

10 35

iii) y = 25

x y

–10 25

0 25

10 25

iv) y = x2 + 25

x y

–10 125

–5 50

0 25

5 50

10 125

b) i) The graphs in parts i, ii, and iii are straight lines, so their equations represent linear

relations; that is, x = –2, y = x + 25, and y = 25.

The graph in part iv is not a straight line, so its equation does not represent

a linear relation.




16

Master 5.12

+10

+1

+1

+1

+10

+10

+3

+5

+7

+9

+1

+1

+1

+1

3. Which relation is linear? Justify your answer.

a) A dogsled moves at an average speed of 10 km/h along a frozen river.

The distance travelled is related to time.

b) The area of a square is related to the side length of the square.

Solutions

Create a table of values, then check to see if the relation is linear.

a) Every hour, the distance travelled by the dogsled increases by 10 km.

There is a constant change of 1 in the 1st column and a constant change of 10 in the

2nd column, so the relation is linear.

b) The area of a square is the square of its side length.

There is a constant change of 1 in the 1st column, but the differences in the 2nd column

are not constant. So, the relation is not linear.

Homework pg 307

#3-5

#6-10, 17

#18, 19,22

Time (hours) Distance

Travelled (km)

0 0

1 10

2 20

3 30

Side Length

(cm)

Area (cm2)

1 1

2 4

3 9

4 16

5 25




17

Master 5.12

4. A hot tub contains 1600 L of water.

Graph A represents the hot tub being filled at a constant rate.

Graph B represents the hot tub being emptied at a constant rate.

a) Identify the dependent and independent variables.

b) Determine the rate of change of each relation, then describe what it represents.

Solution

For Graph A

a) The dependent variable is the volume, V. The independent variable is the time, t.

b) Choose two points on the line.

Calculate the change in each variable from one point to the other.

Change in volume: 800 L – 0 L = 800 L

Change in time: 40 min – 0 min = 40 min

Rate of change: 800 L

40 min = 20 L/min

The rate of change is positive so the volume

is increasing with time. Every minute,

20 L of water are added to the tank.

For Graph B

a) The dependent variable is the volume, V. The independent variable is the time, t.

b) Choose two points on the line.

Calculate the change in each variable from one point to the other.

Change in volume: 0 L – 1600 L = –1600 L

Change in time: 40 min – 0 min = 40 min

Rate of change: 1600 L

40 min

= –40 L/min

The rate of change is negative so the volume

is decreasing with time. Every minute,

40 L of water are removed from the tank.




18

Master 5.13

5.7 Graphs of Linear Functions pg 311-318 outcome R5

1. This graph shows how the height of a burning candle changes with time.

a) Write the coordinates of the points where the graph intersects the axes. Determine the

vertical and horizontal intercepts. Describe what the points of intersection represent.

b) What are the domain and range of this function?

Solution

a) On the vertical axis, the point of intersection has coordinates (0, 10). The vertical intercept

is 10. This point of intersection represents the height of the candle when the time is 0 min;

that is, the height of the candle before it is lit: 10 cm

On the horizontal axis, the point of intersection has coordinates (45, 0). The horizontal

intercept is 45. This point of intersection is the time taken until the height of the candle is 0

cm; that is, the time it takes for the candle to burn completely: 45 min

b) The domain is the set of possible values of time: 0 ≤ t ≤ 45

The range is the set of possible values of the height of the candle: 0 ≤ h ≤ 10

2. Sketch a graph of the linear function f(x) = 4x – 3.

Solution

f(x) = 4x – 3

Since the function is linear, its graph is a straight line.

Determine the y-intercept: Determine the x-intercept:

When x = 0, When f(x) = 0,

f(0) = 4(0) – 3 0 = 4x – 3

f(0) = –3 3 = 4x – 3 + 3

3 = 4x

x = 3

4

Determine the coordinates of a third point on the graph.

When x = 1,

f(1) = 4(1) – 3

f(1) = 1

Plot the points (0, –3), 3

, 04

, and (1, 1), then draw a line through them.




19

Master 5.13

3. Which graph has a rate of change of –5 and a vertical intercept of 100?

Justify your answer.

a) b)

Solution

a) The graph of d = f(t) has a vertical intercept of 100.

The rate of change is: 100

20

= –5

So, it is the correct graph.

b) The graph of d = k(t) has a vertical intercept of 100.

The rate of change is: 100

20 = 5

So, it is not the correct graph.


The right to reproduce or modify this page is restricted to purchasing schools. 23 This page may have been modified from its original. Copyright © 2010 Pearson Canada Inc.

Master 5.13

4. This graph shows the total cost for a house call

by an electrician for up to 6 h work.

The electrician charges $190 to complete a job.

For how many hours did she work?

Solutions

Method 1

To estimate the number of hours worked

when the total cost is $190, use the graph.

From the graph, the electrician worked for about 31

4 h.

Method 2

The cost for a house call is the cost when the number of hours worked is 0.

This is the vertical intercept of the graph, which is 60.

The cost of a house call is $60.

The increase in cost for each additional hour worked is the rate of change of the function.

Determine the change in each variable.

The graph shows that for every 3 h worked,

the cost increases by $120.

The rate of change is: $120

3 h= $40/h

The increase in cost for each additional hour worked is $40.

An equation that represents this situation is: C = 40n + 60

To determine the number of hours worked

when the total cost is $190, use the equation:

C = 40n + 60 Substitute: C = 190

190 = 40n + 60 Solve for n.

190 – 60 = 40n + 60 – 60

130 = 40n

130

40 =

40

40

n

13

4 = n So, the electrician worked for 3

1

4 h.

Homework


21 The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2010 Pearson Education Canada

Master 5.13

Pg 319

#4-5

#6,7,9,11,14,15,17,

#18,19

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