unit 4 relations & functions - mr. kosik's mathtime · unit 4 relations & functions...
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1
Unit 4 Relations & Functions
General Outcome: • Develop algebraic and graphical reasoning through the study of relations.
Specific Outcomes:
4.1 Interpret and explain the relationship among data, graphs, and situations.
4.2 Demonstrate an understanding of relations and functions.
4.3 Describe and represent linear relations, using:
o words
o ordered pairs
o tables of values
o graphs
o equations
4.4 Determine the characteristics of the graphs of linear relations, including the:
• intercepts
• slope
• domain
• range
Topics:
• Representing Relations (Outcome 4.3) Page 2
• Functions (Outcome 4.2) Page 9
• Function Notation (Outcome 4.2) Page 14
• Interpreting & Sketching (Outcome 4.1) Page 27
Graphs
• Continuous & Discrete Data (Outcome 4.1) Page 38
• Vertical Line Test (Outcome 4.2) Page 40
• Non-Linear Relations (Outcome 4.1) Page 41
• Linear Relations (Outcome 4.4) Page 48
• Intercepts (Outcome 4.4) Page 57
2
Unit 4 Relations & Functions
Representing Relations:
A relation links elements from one set to elements of another
set. Relations can be represented in various ways. The first set of
elements is called the Domain (the Independent variable)and
the second set of elements is called the Range (the Dependent
variable).
Table:
Fruit Color x y
Apple Red 4 8
Apple Green 6 12
Banana Yellow 7 14
Strawberry Red 8 16
Cherry Red 11 22
Cherry Yellow 15 30
Blueberry Blue 45 90
Arrow Diagram:
Apple 4 8
Banana Red 6 12
Strawberry Green 7 14
Cherry Yellow 8 16
Blueberry Blue 11 22
15 30
45 90
3
Ordered Pairs:
(Apple, Red), (Apple, Green), ( )4, 8 , ( )6, 12 , ( )7, 14
(Banana, Yellow), ( )8, 16 , ( )11, 22 , ( )15, 30
(Strawberry, Red), (Cherry, Red) ( )45, 90
(Cherry, Yellow), (Blueberry, Blue)
Words:
Apples, strawberries, and The value of y is 2 times
cherries are all red. Apples the value of x.
are also green. Bananas and
cherries are yellow. Blueberries
are blue.
Equation:
Not possible with this relation. 2y x=
Graph:
This relation cannot be
Graphed. (Graphs are
typically used to describe
relations involving
values.)
4
Ex) Different towns in British Columbia can be associated with
the average time, in hours, it takes to drive to Vancouver.
This relation is described by the graph below.
a) Describe this relation using
ordered pairs.
b) Describe this relation c) Describe this relation
using an arrow diagram. using a table.
e) Identify the domain. f) Identify the range.
5
Ex) Consider the relation represented by the ordered pairs.
( )7, 73 , ( )5, 53 , ( )9, 93 , ( )2, 23 , ( )0, 3 , ( )4, 37−
a) Represent the above b) Represent the above
relation as a table of relation in words.
values.
c) Represent the above relation as an equation.
d) Identify the domain. e) Identify the range.
6
Representing Relations Assignment:
1) Given the table below, represent the relation:
Coin Value ($) a) using words
penny 0.01
nickel 0.05
dime 0.10
quarter 0.25
loonie 1.00
toonie 2.00
b) as a set of ordered pairs c) as an arrow diagram
2) Given the table below, represent the relation:
Sport Equipment a) using words
badminton shuttlecock
badminton racquet
hockey puck
hockey stick
tennis ball
tennis racket
soccer ball
b) as a set of ordered pairs c) as an arrow diagram
7
3) Given the table below, represent the relation:
Artist Medium a) using words
Gaetanne sculpture
Hubert painting
Huguette stained glass
James painting
Nathalie photography
Simone photography
b) as a set of ordered pairs c) as an arrow diagram
4) Given the graph below, represent the relation as:
a) a table b) an arrow diagram
8
5) A digital clock displays digits from 0 to 9 by lighting up different segments in
two squares. For example, the digit 2 needs 5 segments to light up, as shown
below.
a) list the set of ordered pairs in the form (digit, number of line segments
required) for this relation.
b) represent this relations using c) represent this relation using
a table of values an arrow diagram
9
Functions:
A function is a special type of relation in which each element in
the domain leads to exactly one element in the range.
Ex) Consider the relation represented by the arrow diagram
below.
January a) Is this relation
February a function?
March
April
May 28
June
July 30
August
September 31
October
November
December
b) Identify the domain.
c) Identify the range.
10
Ex) Identify which of the following relations are functions.
a) ( )4, 10 , ( )5, 12 , ( )8, 30 , ( )4, 7− , ( )12, 51
b) ( )3, 11 , ( )5, 17 , ( )8, 17 , ( )21, 10 , ( )2, 19− −
Ex) Shauna gets paid $22.00 for each hour of work she does.
a) Create a table that b) Describe this
describes this relation. relation using
an equation.
c) Is this relation a
function?
11
Functions Assignment:
1) State whether each relation represented as an arrow diagram is a function.
a) b) c)
2 4 0 2 4
3 6 4 1 2−
4 8 2 3 9
5 10 3 3−
2) For each relation represented as a set of ordered pairs state whether or not it is
a function and then indicate the domain and range.
a) ( ) ( ) ( ) ( ) 1, 3 , 2, 6 , 3, 9 , 4, 12
b) ( ) ( ) ( ) ( ) 1, 0 , 0, 1 , 1, 0 , 0, 1− −
c) ( ) ( ) ( ) ( ) 2, 3 , 4, 5 , 6, 7 , 8, 9
d) ( ) ( ) ( ) ( ) ( ) ( ) 0, 1 , 0, 2 , 1, 2 , 0, 3 , 1, 3 , 2, 3 \
12
3) Answer the following given the table below.
Number of Cans of
Juice Purchased, n
Cost, C
($)
a) Explain why the relation is a
function.
1 2.39
2 4.00
3 6.39
4 8.00
5 10.39
6 12.00
b) Identify the independent and c) State the domain and range.
dependent variable
4) Answer the following given the table below.
Altitude, A
(m)
Temperature, T
( C )
a) Explain why the relation is a
function.
610 15.0
1220 11.1
1830 7.1
2440 3.1
3050 0.8−
3660 4.8−
b) Identify the independent and c) State the domain and range.
dependent variable
13
5) The Rassemblement jeunesse francophone in Alberta brings together French
language high school students from all over the province for a day of activities.
Use two columns in the table below to represent a relation.
Name From Age Gender
Marie Edmonton 13 F
Gabriel Falher 16 M
Elise Bonnyville 14 F
Christopher Calgary 13 M
Jean Edmonton 15 M
Melanie Edmonton 15 F
Nicole Red Deer 17 F
Marc Legal 13 M
a) Name two relations that are functions.
b) Name two relations that are not functions.
14
Function Notation:
Rather than using “ y =” to represent the potential values of a
function, Function Notation is used where different functions are
given different names.
Example:
4 6
becomes
( ) 4 6
y x
f x x
= −
= −
2 3
becomes
g( ) 2 3
y x
x x
= − +
= − +
2
2
11
becomes
( ) 11
y x
h x x
= +
= +
This way we can reference different functions without
confusion. (If I talk about ( )g x , you know which function I am
referring to.)
Think of ( )f x , ( )g x , or ( )h x simply replacing “y”.
Ex) ( ) 4 6f x x= −
This is read as f of x or f at x.
It means that the function is called function f, and that x is
the variable used within the function.
( )f x does not mean that f and x are multiplied together.
Instead it means that the function f is written in general
terms using x.
(3)f would be referring to the value of the function at 3.
(What is y when x is 3?)
15
Ex) If ( ) 3 8f x x= − , determine the following.
(2)f (5)f ( 6)f −
( )23
f ( )3f ( )f
Ex) If g( ) 2x x= + , ( ) 5 7h x x= − and 2( ) 3k x x x= − determine
the following.
a) (4)h b) (0)g c) (3)k
d) (2) (1)k h+ e) (5) (2)g k−
16
Ex) If ( ) 5 6g x x= − + , determine the following.
a) ( )g a b) (7 )g a
c) ( 2)g a + d) ( 3)g x −
Ex) Determine ( 2)h x + if:
a) ( ) 7 2h x x= − b) ( )3
xh x
x=
+ c) ( ) 5h x x= −
Ex) If ( ) 5 6f x x= − + , determine the value of x when
a) ( ) 34f x = − b) ( ) 16f x =
17
Ex) Complete the table of values given below if ( ) 2 4b x x= − .
x ( )b x
3−
1−
0
2
17
Ex) Complete the table of values given below if ( ) 3 5f x x= + ,
( ) 8 4g x x= − + , and 1
( ) 92
h x x= − .
x ( )f x ( )g x ( )h x
6−
4−
0
12
18
Ex) Plot the following on the grid provided.
x ( )k x
5− 9−
4− 7−
1− 1− 0 1
2 5
3 7
4 9
Ex) Given the graph of ( )y m x= , determine the following.
a) (2)m
b) ( 3)m −
c) (5)m
19
Ex) If ( ) 2h x x= + plot the following points.
(6)h
( 7)h −
(1)h
( 11)h −
Ex) Given the table of values below, determine the following.
x ( )p x ( )q x ( )r x
4− 5 7 2
0 11 8− 38
18 4− 19 0
25 0 20− 34
a) (18)p b) ( 4)q − c) (0)r
d) If ( ) 4p x = − , what is x?
20
Function Notation Assignment:
1) Given ( ) 8 3f x x= − , 2( ) 2g x x= + , and 24
( )h xx
= evaluate the following.
a) (7)f b) (5)g c) (3)h
d) ( 4)h − e) ( 3)f − f) ( 6)g −
g) 5
2f
h) (15)h i) 1
3g
j) ( )2 (6)f h+ k) (1) (2) ( 3)g h f + −
21
2) If ( ) 12 11h x x= − determine the following.
a) (4)h b) 3
4h
c) ( 1)h a −
d) (3 )h a e) (2 5)h a +
3) If 2( ) 2f x x= + and ( ) 10g x x= − determine the following.
a) 4 (3)g b) ( 5) 8f − − c) 2 (4) 2 (4)f g+
d) ( )5 ( 3) ( 3)f g− + − e) ( )(7)f g
22
4) If ( ) 6 11m x x= + determine the value of x when
a) ( ) 53m x = b) ( ) 67m x = −
5) If ( ) 43
xk x = + determine the value of x when
a) ( ) 15k x = b) ( ) 21k x = −
6) Determine the following given the table below.
x ( )f x ( )g x ( )h x
9− 45− 3− 0
2− 10− 4 7
0 0 6 9
12 60 18 21
a) ( 2)f − b) (12)h c) (0)g
d) ( 2) ( 9)g h− + − e) (12)
( 9)
h
g − f) 3 ( 2) 4 ( 2)h f− − −
23
7) Determine the following given the information below.
x ( )b x ( ) 3 1c x x= +
5− 27
0 7
1 5
5 13−
8 25−
a) ( 5) (5)b c− + b) (1) (5)c b− c) ( )7 (0) (0)b c+
d) (2)
(1)
c
b e) (8) (8)b c f) (1) (3) (8) (2)b c b c− + −
8) Create an arrow diagram that represents the following function.
(2) 7f = , (5) 11f = , ( 4) 8f − = , (0) 20f = , ( 6) 7f − = , (10) 8f =
24
9) Given the graphs of ( )y f x= and ( )y g x= determine the following.
( )y f x= ( )y g x=
a) (2) (4)f g+ b) (0) (0)g f− c) ( 6) ( 9)f g− −
d) 4 ( 3) (7)g f− + e) ( )3 (8) (8)f g− f) ( 5) (1) (5)f g g− +
g) determine the value of x when
i) ( ) 3g x = ii) ( ) 3f x = iii) ( ) 6f x = −
25
10) Determine the following given the arrow diagrams below.
x ( )f x x ( )g x
3 1− 1− 2−
5 2 2 3
6 4 6 6
8 6 7
a) (5) (6)f g+ b) (2) (6)g f− c) (8)
( 1)
f
g −
d) 3 (7) (6)g f+ e) ( )(7)f g f) ( )(6)g f
g) determine the value of x when
i) ( ) 2f x = ii) ( ) 2g x = − iii) ( ) 3g x =
26
11) Given the graphs of ( )f x and ( )g x , determine the following.
a) (4)f b) ( 8)g − c) ( 3) ( 2)g f− − −
d) 6 (4) 3 (3) 12g f− + e) (4)
(0) (0)( 8)
fg f
g+ −
−
f) ( 2) (5) (8)
( 3)
g f g
f
−
− g)
(4) ( 5)
(3) (2)
g f
f g
−+
27
Interpreting & Sketching Graphs:
By interpreting the information a graph displays we get a story
of what is happening with the relation.
Ex) Answer the following given the graph showing the depth
of a scuba diver as a function of time.
a) How many minutes did the dive last?
b) At what times did the diver stop her descent?
c) What was the greatest depth the diver reached? For
how many minutes was the diver at that depth?
28
Ex) Given below is a graph that illustrates the relationship of
the depth of water in a bathtub as a function of time. What
does each segment of the graph represent?
O to A: A to B:
B to C: C to D:
D to E: E to F:
F to G:
29
Ex) Each point of the graph below represents a bag of popcorn.
a) Which bag is the most
expensive? What does it
cost?
b) Which bag has the least mass? What is this mass?
c) Which bags have the same mass? What is this mass?
d) Which bags cost the same? What is the cost?
e) Which of bags C or D has the better value?
f) If a person were to purchase one of each bag, how
much would it cost?
30
Ex) Given a below, describe the journey for each segment of
the graph .
O to A:
A to B:
B to C:
C to D:
D to E:
31
Ex) Hailey went on a bike ride. She accelerated until she
reached a speed of 20 km/h (this took 122
minutes), then
she cycled for 30 minutes at approximately 20 km/h.
Hailey arrived at the bottom of a hill, and her speed
decreased to approximately 5 km/h (this took 122
minutes). She maintained the speed for 10 minutes as she
cycled up the hill. She stopped at the top of the hill for 10
minutes.
Sketch a graph of speed as a function of time for this
scenario.
32
Interpreting & Sketching Graphs Assignment:
1) Each point on the graph below represents a polar bear.
a) Which bear has the greatest mass and
what is its mass?
b) Which bear is the shortest and what is
its height?
c) Which two bears have the same d) Which two bears have the same
mass and what is this mass? height and what is this height?
2) The graph below shows the height of the tide in a harbor as a function of time
in one day.
a) What is the greatest height and at
what times does it occur?
b) What is the least height and at what
times does it occur?
c) How high is the tide at 04:00? d) When is the tide 4 m high?
33
3) To raise a flag, Sepideh pulls the rope steadily with both hands for a short time,
then moves both hands up the rope and pulls again. She does this until the flag
has been raised. Which graph best represents the height of the flag?
4) Gill runs for exercise. This graph shows her distance from home during one of
her runs. Describe Gill’s run for each segment of the graph.
Segment OA:
Segment AB:
Segment BC: Segment CD:
34
5) Katanya went scuba diving in Egypt. The graph below shows her depth below
sea level as a function of time on one of her dives. Describe all that you know
about the dive from the graph.
Segment OA:
Segment AB: Segment BC:
Segment CD: Segment DE:
35
6) An oven is turned on at a room temperature of 20 C and it takes 10 min. to
reach a temperature of 190 C . A tray of cookies is placed in the oven to bake
for 10 min. The oven is then turned off and returns to room temperature after
15 min. Sketch a graph of temperature as a function of time. Label each section
and explain what it represents.
36
7) Zac goes on a day long snowmobile trip. His snowmobile will hold 30 L of gas
when full. Throughout the day:
• Zac’s snowmobile begins the trip with 25 L of fuel.
• Zac travels for 2 hours and uses 10 L of fuel.
• Zac fills the tank of his snowmobile to full and then travel for another 2
hours burning another 10 L of fuel.
• Zac takes a 2 hour break and then travels for another 3 hours to his
destination. This last leg of the journey uses 15 L of fuel.
• Upon arriving at his destination, Zac fills his snowmobile so that it is full of
fuel.
Sketch a graph of volume of gas as a function of time. Label each section and
explain what it represents.
37
8) A student drew a graph to represent this situation.
“Jonah is watching television. After 3 min. his mom enters the room to ask him
a question. He turns the volume down a bit, answers his mom, then turns the
volume back up. Two min. later, Jonah’s dad turns on the dishwasher so Jonah
gradually turns up the volume. After a further 3 min., a commercial comes on
so Jonah presses the mute button.”
Describe any errors in the student’s graph.
38
Continuous & Discrete Data:
Ex) ( )1, 15 , ( )2, 30 , ( )3, 45 , ( )4, 60 , ( )5, 75
a) Plot the above data on b) Plot the above data on
the grid below if the the grid below if the
domain represents the domain represents the
number of people time in seconds and the
attending a movie and range represents the
the range represents the distance a car travels in
cost. metres.
39
Discrete Data:
Simply plotting data on a grid is known as a Scatter Plot, this is
done when data is discrete.
Continuous Data:
A straight line drawn as close as possible to as many points as
possible is called a Line of Best Fit, this is done when data is
continuous.
Ex) Graph 2 3y x= − for a Ex) Graph 2 1y x= − + for a
domain given by domain given by x R .
3, 2, 1, 0, 1− .
40
Vertical Line Test:
If it is possible to draw a vertical line through a graph so that the
line goes through the graph more than once, it is not a function.
(A relation is a function when each value in the domain results
in only one value in the range.)
Ex) Determine whether or not the relations given below are
functions.
41
Non-Linear Relations:
Non-Linear Relations produce graphs that do not form a straight
line.
Ex) Graph each of the following on your calculator, then state
the domain and range.
a) 2y x= b) ( )3
3 2y x= − +
c) 2
1y
x= d) 3xy =
e) 4y x= − f) 3
5
xy =
42
Continuous & Discrete Data / Vertical Line Test Assignment:
1) State the domain and range for each function given below.
a) b) c)
Domain: Domain: Domain:
Range: Range: Range:
2) State whether or not each of the following graph represent a function. Justify
your answer.
a) b)
43
3) Match the graph of each function to its domain and range.
i) Domain: 1 3x ii) Domain: 1 3x
Range: 2 4y Range: 1 4y
iii) Domain: 0x iv) Domain:1 4x
Range: 2y = Range: 1 2y
a)
b)
c)
d)
3) State whether or not each of the following graph represent a function. Justify
your answer.
a) b)
44
c) d)
4) State the domain and range for each function given below.
a)
b)
c)
Domain:
Range:
Domain:
Range:
Domain:
Range:
45
5) Consider the graphs given below.
a) For each graph identify the independent and dependent variables.
b) Why are the points connected on one graph but not on the other?
46
6) Paulatuuq is north of the Arctic Circle. The table below shows the number of
hours, h, the sun is above the horizon every 60 days from January 1st, which is
day 0.
Day h a) Identify the independent and dependent variable.
b) If this data was presented as a graph would the points
be connected? Explain your reasoning.
0 0
60 9.7
120 18.5
180 24.0
240 15.9
300 7.4
360 0
7) Latex paint is sold in 4 L cans. Each can costs $40 and will cover 34 m2.
a) Complete the table given below.
Volume of Paint (L) 0 4 8 12 16
Cost ($) 0 40
Area Covered (m2) 0 34
b) If a graph of cost as a function of volume was created, should the point be
connected? Explain your reasoning.
c) If a graph of area covered as a function of volume was created, should the
points be connected? Explain your reasoning.
47
8) Answer the following given the graphs below.
a)
b)
i) Determine the range value
when the domain value is 0
ii) Determine the domain value
when the range value is 5
i) Determine the range value
when the domain value is 2−
ii) Determine the domain value
when the range value is 0
48
Linear Relations:
Linear Relations produce graphs that are straight, unbroken
lines.
For a linear relation, a constant change in the independent
variable (x-variable or domain) results in a constant change in
the dependent variable (y-variable or range).
Ex) The cost for a rental car is $60, plus $20 for every 100 km
driven. The table below shows this relation between the
dependent and independent variables.
Distance (km) Cost ($) a) Is this an example
of a linear relation? 0 60
100 80
200 100
300 120
400 140
b) Graph the relation below. c) Determine the Rate
of Change.
49
Ex) For each case below, determine whether or not the relation
is linear and if so determine its rate of change.
a)
x y
0 32
5 41
10 50
15 59
20 68
b)
x y
0 0
5 75
10 300
15 675
20 1200
c)
x y
45 94
52 122
59 150
66 178
73 206
50
Ex) A water tank on a farm holds 6000 L. Graph A represents
the tank being filled at a constant rate. Graph B represents
the tank being emptied at a constant rate.
a) Identify the independent and dependent variables.
b) Determine the rate of change of each relation, then
describe what it represents.
51
Linear Relations Assignment:
1) Determine which tables represent linear relations.
a) b)
Time
(min)
Distance
(m)
Time
(s)
Speed
(m/s)
0 10 0 10
2 50 1 20
4 90 2 40
6 130 3 80
c) d)
Speed
(m/s)
Time
(s)
Distance
(m)
Speed
(m/s)
15 7.5 4 2
10 5 16 4
5 2.5 1 1
0 0 9 3
2) Determine which sets of ordered pairs represent a linear function.
a) ( ) ( ) ( ) ( ) 3, 11 , 5, 9 , 7, 7 , 9, 5
b) ( ) ( ) ( ) ( ) 2, 3 , 0, 1 , 2, 3 , 4, 7− − −
c) ( ) ( ) ( ) ( ) 1, 1 , 1, 3 , 2, 1 , 2, 3
52
3) Graph each relation given below. Create a table of values if necessary.
a) 2 8y x= +
b) 0.5 12y x= +
53
c) 7x =
d) 6x y+ =
54
4) The distance required for a car to come to a complete stop after its breaks are
applied is the braking distance. The breaking distance, d, in metres, is related
to the speed of the car, s kilometers per hour, when the brakes are first applied.
s (km/h) d (m) a) Identify the dependent and independent variables
50 13
60 20
70 27
80 35
b) Determine whether the relation is linear. If the relation is linear, determine
its rate of change.
5) The altitude of a plane, a metres, is related to the time, t minutes, that has
elapsed since it started its descent.
t (min) a (m) a) Identify the dependent and independent variables.
0 12 000
2 11 600
4 11 200
6 10 800
8 10 400
b) Determine whether the relation is linear. If the relation is linear, determine
its rate of change.
55
6) Earth rotates through approximately 360 C every 24 h. The set of ordered
pairs below describes the rotation. The first coordinate is the time in hours, and
the second coordinate is the approximate angle of rotation in degree.
Determine if this relation is linear. If the relation is linear determine its rate of
change.
( ) ( ) ( ) ( ) ( ) 0, 0 , 6, 90 , 12, 180 , 18, 270 , 24, 360
7) Sophie and 4 of her friends plan a trip to the Edmonton Chante for one night.
The hotel room is $95 for the first 2 people, plus $10 for each additional
person in the room. The total cost is related to the number of people. Is the
relation linear? If the relation is linear determine its rate of change.
8) A skydiver jumps from an altitude of 3600 m. For the first 12 s, her height in
metres above the ground is described by this set of ordered pairs:
( ) ( ) ( ) ( ) 0, 3600 , 4, 3526 , 8, 3353.5 , 12, 3147.5
For the next 21 s, her height above the ground is described by this set of
ordered pairs:
( ) ( ) ( ) ( ) 15, 2988.5 , 21, 2670.5 , 27, 2352.5 , 33, 2034.5
Determine whether either set of ordered pairs represents a linear relation. If the
relation is linear determine its rate of change.
56
9) The graph below represents Jerome’s long distance phone call to his pen pal in
Nunavut. Jerome is charged a constant rate.
a) Identify the dependent and independent
variables.
b) Determine the rate of change and describe
what it represents.
10) Match each description of a linear relation with its equation and set of ordered
pairs below.
Equation 1: 500 40y x= +
Equation 2: 35 0.06y x= −
Equation 3: 20y x=
Set A: ( ) ( ) ( ) ( ) 100, 29 , 200, 23 , 300, 17 , 400, 11
Set B: ( ) ( ) ( ) ( ) 1, 20 , 5, 100 , 10, 200 , 15, 300
Set C: ( ) ( ) ( ) ( ) 0, 500 , 40, 2100 , 80, 3700 , 100, 4500
a) The amount a person earns is related to her hourly wage.
b) The cost of a banquet is related to a flat fee plus an amount for each person
who attends.
c) The volume of gas in a car’s gas tank is related to the distance driven since
the time when the tank was filled.
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Intercepts:
The Vertical or y-intercept is the point where a graph crosses
the vertical or y-axis. The Horizontal or x-intercept is the point
where a graph crosses the horizontal or x-axis.
Ex) For each of the following determine the x and y-intercepts.
a) 2 14y x= − + b) 4 5 60x y− =
c) 2 4 64x y+ = d) 2 49y x= −
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Ex) Given the relation ( ) 4 68f x x= − determine the following.
a) (0)f or y-intercept b) x when ( ) 0f x =
or x-intercept
c) ( 5)f − d) x when ( ) 20f x =
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Ex) The graph below shows the fuel consumption of a scooter
with a full tank of gas at the beginning of a journey.
a) Determine the vertical
intercept and explain
what it represents.
b) Determine the horizontal intercept and explain what it
represents.
c) What are the domain and range of this function?
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Intercepts Assignment:
1) Each graph below shows distance, d kilometres, as a function of time, t hours.
a)
b)
i) Determine the coordinates of
the intercepts.
ii) Determine the rate of change.
iii) Determine the domain and
range
i) Determine the coordinate of
the intercepts.
ii) Determine the rate of change.
iii) Determine the domain and
range
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2) Each graph show the altitude, A feet, of a small plane as a function of time, t
minutes.
a)
b)
i) Determine the coordinates of
the vertical intercept.
ii) Determine the rate of change.
iii) Determine the domain and
range.
i) Determine the coordinates of
the vertical intercept.
ii) Determine the rate of change.
iii) Determine the domain and
range.
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3) The graphs below show the temperature, T degrees Celsius, as a function of
time, t hours, at different locations.
i) ii)
iii) iv)
a) Which graph has a rate of change 5 C /h and a vertical intercept of 10 C− ?
b) Which graph has a rate of change of 10 C− /h and a vertical intercept of 20 C ?
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4) The graph below shows the labour cost for running a backhoe.
a) Determine the coordinates of the
point where the graph intersects
the axes. What does this
represent?
b) Determine the rate of change. What does this represent?
c) State the domain and range.
d) What is the cost to run the backhoe for 7 h?
e) For how many hours is the backhoe run when the cost is $360?
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5) The graph below shows the cost for a cab. The cost, C dollars is a function of
the distance travelled, d kilometres.
a) Determine the rate of change. What
does this represent?
b) What is the cost when the c) What is the distance when the
distance is 7 km? cost is $9.50?
6) The graph below shows the distance to the finish line, d kilometres, as a
function of time, t hours, for one dogsled in a race near Churchill, Manitoba.
a) What was the length of time it took the dogsled
to finish the race?
b) What was the average speed of the dogsled?
c) How long was the race in d) What time did it take for the
kilometres? Dogsled to complete 2
3 of the race?
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7) The capacity of each of 2 fuel storage tanks is 100 m3. Graph A represents the
volume of fuel in one tank as a function of time as the tank is filled. Graph B
represents the volume of fuel in another tank as a function of time as the tank
is emptied.
a) Does it take longer to fill the empty tank or empty the tank?
b) In the time it takes for one tank to be half empty, about how much fuel
would be in a tank that was being filled from empty?
66
8) Northlands School Outdoor Club had a fundraiser to help purchase snowshoes.
The club had 300 power bars to sell. The graph below shows the profit made
from selling power bars.
a) Determine the rate of change. What does
this represent?
b) Determine the coordinates of the intercepts. What does each one represent?
c) State the domain and range.
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Answers
Representing Relations Assignment:
1. a) A penny has a value of $0.01, a nickel has a value of $0.05, a dime has a
value of $0.05, a quarter has a value of $0.25, a loonie has a value of $1.00,
and a toonie has a value of $2.00.
b) c)
(penny, 0.01) dime 0.01
(nickel, 0.05) loonie 0.05
(dime, 0.10) nickel 0.10
(quarter, 0.25) penny 0.25
(loonie, 1.00) quarter 1.00
(toonie, 2.00) toonie 2.00
2. a) Some of the equipment required to play badminton are a shuttlecock and a
racquet, for hockey a puck and stick are required, for tennis some of the
equipment includes a ball and a racket, and for soccer you will require a
ball.
b) c)
(badminton, shuttlecock) badminton ball
(badminton, racquet) hockey puck
(hockey, puck) soccer racquet
(hockey, stick) tennis shuttlecock
(tennis, ball) stick
(tennis, racquet)
(soccer, ball)
3. a) Gaetanne is a sculpture artist, Hubert and James are painters, Huguette
creates stained glass, and Nathalie and Simone are photographers.
b) c)
(Gaetanne, sculpture) Gaetanne
(Hubert, painting) Hubert painting
(Huguette, stained glass) Huguette photography
(James, painting) James sculpture
(Nathalie, photography) Nathalie stained glass
(Simone, photography) Simone
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4. a) b)
Word Number of
Letters
Blue 3
Green 4
Orange 5
Red 6
Yellow
Blue 4
Green 5
Orange 6
Red 3
Yellow 6
5. a) (0, 6), (1, 2), (2, 5), (3, 5), (4, 4), (5, 5), (6, 5), (7, 3), (8, 7), (9, 5)
b) b)
Digit Number of
Segments
0
1
2 2
3 3
4 4
5 5
6 6
7 7
8
9
0 6
1 2
2 5
3 5
4 4
5 5
6 5
7 3
8 7
9 5
Functions Assignment:
1. a) function b) not a function (relation) c) function
2. a) function Domain: 1, 2, 3, 4 Range: 3, 6, 9, 12
b) not a function Domain: 1, 0, 1− Range: 1, 0, 1−
c) function Doman: 2, 4, 6, 8 Range: 3, 5, 7, 9
d) not a function Domain: 0, 1, 2 Range: 1, 2, 3
3. a) Each item in the domain (number of cans) results in one range value (cost).
b) Independent Variable: Number of Cans of Juice Purchased
Dependent Variable: Cost
c) Domain: 1, 2, 3, 4 ,5, 6 Range: 2.39, 4.00, 6.39, 8.00, 10.39, 12.00
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4. a) Each altitude (item in the domain) results in one temperature (item in the
range).
b) Independent Variable: Altitude Dependent Variable: Temperature
c) Domain: 610, 1220, 1830, 2440, 3050, 3660
Range: 4.8, 0.8, 3.1, 7.1, 11.1, 15.0− −
5. a) (Name, From), (Name, Age), (Name, Gender)
b) (From, Name), (From, Age), (From, Gender), (Age, Name), (Age, From)
(Age, Gender), (Gender, Name), (Gender, From), (Gender, Age)
Function Notation Assignment:
1. a) 53 b) 27 c) 8 d) 6− e) 27− f) 38 g) 17 h) 8
5
i) 19
9 j) 17 k) 9
2. a) 37 b) 2− c) 12 23a − d) 36 11a − e) 24 49a +
3. a) 28 b) 19 c) 48 d) 120 e) 11
4. a) 7x = b) 13x = −
5. a) 33x = b) 75x = −
6. a) 10− b) 21 c) 6 d) 4 e) 7− f) 61
7. a) 43 b) 17 c) 56 d) 7
5 e) 625− f) 37−
8. 6−
4− 7
0 8
2 11
5 20
10
9. a) 8 b) 2 c) 27− d) 30 e) 48 f) 9−
g) i) 4x = ii) 0x = iii) 3x = −
10. a) 12 b) 1 c) 2− d) 11 e) 1− f) 3
g) i) 6x = ii) 1x = − iii) 2x = or 7x =
11. a) 8 b) 3− c) 6 d) 0 e) 7
3 g) 15 h)
19
6
−
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Interpreting & Sketching Graphs Assignment:
1. a) Bear F; approximately 650 kg b) Bear A; approximately 0.7 m
c) Bears D and F; 400 kg d) Bears D & H; approximately 2.25 m
2. a) 8 m at 06:00 and 18:00 b) 2 m at 00:00, 12:00, and 24:00
c) approximately 02:00, 09:40, 14:20, and 21:40
3. Graph B
4. Segment OA: Gill runs 1 km in 5 minutes away from her home.
Segment AB: Gill rests for 5 minutes.\
Segment BC: Gill runs another 1 km farther away from her home in 10 minutes
(slower than before).
Segment CD: Gill turns around and runs back home, a distance of 2 km in 10
minutes.
5. Segment OA: Katanya descends 15 m in 4 minutes.
Segment AB: Katanya remains at a depth of 15 m for 6 minutes.
Segment BC: Katanya descends another 10 m in 4 minutes.
Segment CD: Katanya remains at a depth of 25 m for 4 minutes.
Segment DE: Katanya ascends to the surface (a distance of 25 m) in 10 minutes.
6.
Segment AB: Oven is warming up from room
temperature of 20 C to 190 C (this takes
10 min.)
Segment BC: The cookies are in the oven baking
cooking for 10 min.
Segment CD: The oven is turned off an cools to
room temperature (this takes 15 min.)
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7.
Segment AB: Zac travel for 2 hours using
10 L of fuel.
Segment BC: Zac fills his snowmobile with
15 L of fuel so it is full.
Segment CD: Zac travels for another 2
hours using 10 L of full.
Segment DE: Zac rests for 2 hours.
Segment EF: Zac travels for another 3
Hours using 15 L of full.
Segment FG: Zac has completed his journey
and refills his snowmobile with gas
so that it is full.
8. From 3 min. to 4 min. the volume should be below 40 because Jonah turns the
volume down.
Continuous & Discrete Data / Vertical Line Test Assignment:
1. a) Domain: 2, 1, 0, 1, 2− − Range: 4, 2, 0, 2, 4− −
b) Domain: 3, 1, 0, 2, 3− − Range: 2, 0, 1, 2, 3−
c) Domain: 3, 2, 1, 0, 1, 2, 3− − − Range: 2
2. a) This is a function as it passes the vertical line test.
b) This is not a function as it fail the vertical line test.
3. a) Function, passes vertical line test
b) not a function, fails vertical line test when 1x =
c) not a function, fails vertical line test when 5x = ( 2 or 3y = )
d) not a function, fails vertical line test for 0 2x
4. a) Domain: 1x Range: y R b) Domain: 3 3x− Range: 3 0y−
c) Domain: 1 2x− Range: 0 3y
5. a) Graph A: Independent Var.: Time Dependent Var.: Distance from School
b) Graph B: Independent Var.: Time Dependent Var.: Number of Students
6. a) Independent Variable: Day Dependent Variable: Hours of Sunlight
b) The point would be connected as data points will exist between those that
are plotted. It is possible to have a partial number of daylight hours.
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7. a)
Volume of Paint (L) 0 4 8 12 16
Cost ($) 0 40 80 120 160
Area Covered (m2) 0 34 68 102 136
b) The points should not be connected as the data is discrete, it is not possible
to purchase a partial can of paint.
c) The points can be connected as the data is continuous, a full can of paint
does not have to be used when painting.
8. a) i) 1− ii) 3 b) i) 5 ii) 3
Linear Relations Assignment:
1. a) Linear Relation b) Not a Linear Relation c) Linear Relation
d) Not a Linear Relation
2. a) Linear Relation b) Linear Relation c) Not a Linear Relation
3. a) b)
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c) d)
4. a) Independent Variable: Speed of Car Dependent Variable: Breaking Distance
b) Not Linear
5. a) Independent Variable: Time Dependent Variable: Altitude
b) Linear; Rate of Change = 400− m/min
6. Linear Relation; Rate of Change = 15 C /h
7. Linear Relation; Rate of Change = $10 /person
8. first set: Not Linear
second set: Linear Relation; Rate of Change = 318− m/s
9. a) Independent Variable: Time Dependent Variable: Cost
b) Rate of Change = $0.08/min
10. a) Equation 3, Set B b) Equation 1, Set C c) Equation 2, Set A
Intercepts Assignment:
1. a) i) Horizontal Intercept: ( )0, 0 Vertical Intercept: ( )0, 0
ii) Rate of Change = 40 km/h iii) Domain: 0 3t Range: 0 120d
b) i) Horizontal Intercept: ( )4, 0 Vertical Intercept: ( )0, 100
ii) Rate of Change = 25− km/h iii) Domain 0 4t Range: 0 100d
2. a) i) ( )0, 400 ii) Rate of Change = 100 ft/min
iii) Domain: 0 8t Range: 400 1200A
b) i) ( )0, 1000 ii) Rate of Change = 50− km/h
iii) Domain 0 8t Range: 600 1000A
74
3. a) Graph ii b) Graph iii
4. a) ( )0, 0
b) $80/h, This represents the situation where if 0 hours are worked, then cost of
running the backhoe is $0.
c) Domain: 0 10t Range: 0 800C d) $560 e) 4.5 hours
5. a) $1.5/km, this represents the cost per kilometre b) $14 c) 4 km
6. a) 2.5 hours b) 24 km/h c) 60 km d) 1 hour 40 min
7. a) It takes longer to fill the tank as this takes 50 minutes and it takes 25 minutes
to empty. b) 25 m3
8. a) $0.80 /bar, this is the profit on each bar sold
b) Vertical Intercept: ( )0, 40− , this represents the start up cost. Before any bars
are sold the club has already spent $40.
Horizontal Intercept: ( )50, 0 , this represents the break even point. The club
needs to sell 50 bars to break even. Any bars sold past this point will make the
club a profit.
c) Domain: 0 300b Range: 40 200P−