LINEAR FUNCTIONS
NOTES 23
There are 3 types of linear functions



Zero variation linear function
Direct variation linear function
Partial variation linear function
How you initially learned the variations:
ZERO VARIATION
DIRECT VARIATION
PARTIAL VARIATION
Now, we are going to look at the linear functions in the following ways:
ZERO VARIATION LINEAR
FUNCTION
DIRECT VARIATION LINEAR
FUNCTION
PARTIAL VARIATION LINEAR
FUNCTION
The rule is:
𝑦=𝑏
The rule is:
𝑦 = 𝑎𝑥
The rule is:
𝑦 = 𝑎𝑥 + 𝑏
Where b = initial value; y
intercept
Where a = slope; rate of change
Where a = slope; rate of change
and
b = initial value; y intercept.
For example:
 Taking the bus costs $3.00
regardless of the distance
travelled.
 Ind var:
 Dep var:
 The rule is:



The graph will pass through the
point (0,3) on the y-axis (this is
your INITIAL VALUE / YINTERCEPT)
The table of values will be:
The graph will look like:
For example:
 A plumber charges $40 per
hour
 Ind var:
 Dep var:
 The rule is:



The graph will pass through
(0,0) and will rise at a rate of
$40 per hour. (this is your
RATE OF CHANGE).
The table of values will be:
For example:
 A plumber charges $20 to
come to your house plus an
additional $30 per hour.
 Ind var:
 Dep var:
 The rule is:


The graph will start on the
y-axis at (0,20). This is your
INITIAL VALUE / YINTERCEPT.
The table of values will be:

The graph will look like:
The graph will look like:
LINEAR FUNCTIONS
FINDING THE RULE OF A LINEAR FUNCTION
NOTES 23
We will start by finding the rule from 2 points (this can be done from 2 points given, picking 2 points
off a graph, or picking 2 points from a table of values). We are going to use points A(5,8) and B(11, 10)
1. Follow the steps for finding the slope
2. When you are finding the rule of a line, we are going to “assume” it’s a partial variation linear
function. It will only be at the end that we can decide which type of function it is. So, we will use
the rule 𝑦 = 𝑎𝑥 + 𝑏. If the y coordinates are the same, you will see that the rule will be a zero
variation linear function. If the x coordinates are the same, your slope will be undefined. This
will be your answer and you WILL NOT go further to find the rule. The rule is x = ____
(whatever those x coordinates were)
3. Plug the slope into the rule
4. Pick one of the points and plug the x and y coordinates into the rule to solve for b.
5. Replace the value of b into the rule.
6. If the value of b is 0, then you know it is direct variation linear function. If it is not, then it is a
partial variation linear function. Write the rule.
LINEAR FUNCTIONS
NOTES 23
DETERMINE THE TYPE OF LINEAR
FUNCTION BASED ON A GRAPH
DETERMINE THE TYPE OF LINEAR
FUNCTION BASED ON A RULE:
1.
2.
3.
4.
𝑦 = 2𝑥 + 6
𝑦=3
𝑦 = −3𝑥
1
𝑦 = −2𝑥 + 0
DETERMINE THE TYPE OF LINEAR FUNCTION BASED ON A TABLE:
x
y
1
5
2
5
3
5
4
5
x
y
1
5
2
8
3
11
4
14
x
y
1
8
3
24
5
40
7
56