Relations and FunctionsLesson 4: Properties of Linear Relations

Todays ObjectivesGraph a set of data and determine

the restrictions on the domain and range

Sort a set of graphs as functions and non-functions

Properties of Linear Relations

Before we start, work with a partner to complete A - C in the Try This activity on page 301. You can use the grid paper on your handout.

Solutions Width (units

)

Area (cm2)

1 2

2 6

3 12

4 20

5 30

6 42

Width (units)

Perimeter

(units)

1 6

2 10

3 14

4 18

5 22

6 26

Non-linear(points not a straight line)

Linear(points in a straight line)

Linear: means a graph has points that connect in a straight line or a graph is a straight line

Linear Relations The cost for a car rental is $60, plus $20 for

every 100 km driven. The independent variable is the distance driven and the dependent variable is the cost.

There are many different ways that we can identify that this is a linear relation: Table of values Set of ordered pairs Graph

Identifying Linear Relations: Table of Values

 Distance (km) Cost ($)

0 60100 80200 100300 120400 140

Constant difference in

independent and dependent variable

= linear relation

For a linear relation, a constant change in the independent variable results in a constant change in the dependent variable.

Identifying Linear Relations: Set of Ordered Pairs

 Again, there is a constant change in the values of the independent variable (first number in each pair) AND in the values of the dependent variable (the second number in each pair

Identifying Linear Relations: Graph Car Rental Cost

Cost ($)

Distance (km)

We can use each representation above to calculate the rate of change. The rate of change can be expressed as a fraction: 

= = $0.20/km

Rate of Change The rate of change is $0.20/km; that is, for

each additional 1 km driven, the rental cost will increase by 20 cents. The rate of change is constant for a linear relation.

In any equation of the form y = mx + b, we can determine the rate of change by looking at the value of m. For example, an equation for this relation could be C = 0.20d + 60, where: C is the dependent variable, cost d is the independent variable, distance, 60 is the initial amount, and 0.20 is the rate of

change (slope)

Example (You do) Graph each equation and state whether or not

it is linear A) y = -3x + 25

B) y = 2x2 + 5

x y

-2 31

-1 28

0 25

1 22

2 19

x y

-2 13

-1 7

0 5

1 7

2 13

SolutionsA B

Linear Non-linear

Example 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.

Graph A: Filling the tank

Volu

me (

L)

Time (min)

a) What is the independent/dependent variable?

b) What is the rate of change?

Example The independent variable is always plotted on

the x-axis, this is t, time. The dependent variable is the volume of water, v.

To find the rate of change it is the same as finding the slope of the line.

Slope = It is best to choose points that are easily read

from the graph to calculate the slope Change in volume:4000L – 3000L = 1000L Change in time: 80 min – 60 min = 20 min Rate of change: The rate of change is positive so the volume is

increasing with time. Every minute, 50L of water are added to the tank.

Example: You do

a) What is the independent/dependent variable?

b) What is the rate of change?V

olu

me (

L)

Time (min)

Graph B: Emptying the Tank

ExampleRate of change: The rate of change is negative so

the volume is decreasing with time. Every minute, 100 L of water are removed from the tank.

HomeworkPg. 308-310

#3,5,7,9,11,13,15,17,19,22 Chapter 5 Vocab Quiz – Next

Wednesday (may include any words from handouts)

Provincial Exam Practice – Next Friday

Chapter 5 Test – Tuesday, March 19

Wall Quiz In teams of 3, move around the

classroom and answer the questions posted on the walls

After the time runs out, return to your seats and we will compare answers

The team(s) with the most correct answers will win candy!