# Day 7 – Characteristics of Linear Relations

```Day 7 – Characteristics of Linear Relations
Minds On
A car rental company charges \$50 per day to rent a mid sized car
plus an administrative fee of \$40 for reserving the car.
This statement is a representation of the relation using words.
Represent the relation using a table, a graph and an equation.
Table:
Graph:
Days (d)
0
1
2
3
4
5
Cost (C)
300
250
200
150
100
50
2
Equation:
4
6
Part A: Write an equation when a relation is represented in words
The equation for a linear relation:
y  mx  b
Considering that a line is really just a set of ordered pairs, (x , y), it makes sense that the equation of a line needs to
contain the variables x and y. These variables will define the coordinates that make up the line.
This means that the only 2 values that need to be determined in order to
write the equation of a linear relation are
m and b.
When a linear relation is represented in words m is the rate of change and b is the initial value.
Linear relation represented in words: m = rate of change
b = initial value
Example 1:
Write an equation for the following relationship by first identifying the value of m and b.
The Copy Centre charges \$75 to design a poster and 25 cents for each copy.
m=
b=
And the equation of this linear relation is:
If The Copy Centre changed their cost per flyer to 35 cents for each copy the equation would become:
If The Copy Centre changed their design cost to \$125 the equation will become:
Part B: Write an equation when a relation is represented in a table of values
The equation for a linear relation:
y  mx  b
m
The rate of change can be determined using a table with:
y
x
The initial value can be found in the table when the independent variable ”x” is zero.
b = the value of y when x is set to zero
Example 2:
a)
Determine the equation of the following linear relations using the tables provided:
x
0
1
2
3
4
y
6
9
12
15
18
b)
x
-2
0
2
4
6
m=
m=
b=
b=
Equation:
Equation:
y
3
5
7
9
11
What should we do if the initial value isn’t in the table?
Example 3:
c)
Determine the equation of the following linear relations using the tables provided:
x
y
2
3
4
-11
-8
-5
d)
x
-8
-6
-4
m=
m=
b=
b=
Equation:
Equation:
y
-5
-10
-15
Part C: Write an equation when a relation is represented as a graph
y  mx  b
The equation for a linear relation:
The rate of change can be determined on a graph with: m 
rise
run
The initial value can be found on the graph by finding the value of the dependent variable “y” when x is zero, this value
occurs when:
b = value of y when the line crosses the y-axis (y-intercept)
Example 4:
Determine the equation of each of the following linear relations.
a)
6
b)
6
y
4
4
2
2
y
x
-5
-5
5
-2
-2
-4
-4
m=
m=
b=
b=
Equation:
Consolidate:
x
5
Equation:
Complete the following table to show how to find the equation of a line using different
representations.
When represented in words
When represented in a table
When represented as a graph
To find
m
To find
b
Replace the m and b with their values in the general equation of a linear relation:
y  mx  b
```