3.2
Linear Functions
How can you determine whether a function is linear or nonlinear?
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
• A linear equation in two variables, x and y, is an equation that can be
written in the form 𝑦 = 𝑚𝑥 + 𝑏, where m and b are constants (just numbers).
• Likewise, a linear function is a function whose graph is a non-vertical line.
• A linear function has a constant rate of change and can be represented by a
linear equation in two variables.
• A nonlinear function does not have a constant rate of change and its graph is
not a line.
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the graph represent a linear or nonlinear function? Explain.
The function is nonlinear.
The function is linear.
Because the graph is not
a line.
Because the graph is a
line.
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the table represent a linear or nonlinear function? Explain.
A linear function has a constant rate of change
+3
+3
+3
As x increases by 3, y decreases by 6.
x
3
6
9
12
y
36
30
24
18
-6
-6
The rate of change is constant for both
x and y.
-6
The function is linear.
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the table represent a linear or nonlinear function? Explain.
A linear function has a constant rate of change
+2
+2
+2
x
1
3
5
7
y
2
9
20
35
+7
+11
+15
As x increases by 2, y increases by
different amounts.
The rate of change is not constant for
both x and y.
The function is nonlinear.
You try!!
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the graph represent a linear or nonlinear function? Explain.
The function is linear.
The function is nonlinear.
Because the graph is a
line.
Because the graph is not
a line.
You try!!
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the table represent a linear or nonlinear function? Explain.
A linear function has a constant rate of change
+1
+1
+1
As x increases by 1, y increases by 2.
x
0
1
2
3
y
3
5
7
9
+2
+2
The rate of change is constant for both
x and y.
+2
The function is linear.
You try!!
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the table represent a linear or nonlinear function? Explain.
A linear function has a constant rate of change
+1
+1
+1
X
1
2
3
4
y
16
8
4
2
-8
-4
-2
As x increases by 1, y decreases by
different amounts.
The rate of change is not constant for
both x and y.
The function is nonlinear.
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Which of the following equations represent linear functions? Explain.
A linear function can be written in the form 𝑦 = 𝑚𝑥 + 𝑏, where m and b are
constants (just numbers).
𝑦 = 3.8,
𝑦 = 𝑥,
𝑦 = 3𝑥 ,
2
𝑦= ,
𝑥
𝑦 = 6(𝑥 − 1),
𝑦 = 3𝑥 ,
You cannot rewrite the equations 𝑦 = 𝑥,
𝑥 2 − 𝑦 = 0 in the form 𝑦 = 𝑚𝑥 + 𝑏.
So these equations cannot represent linear functions.
You can rewrite the equation 𝑦 = 3.8 as 𝑦 = 0𝑥 + 3.8
You can rewrite the equation 𝑦 = 6(𝑥 − 1) as 𝑦 = 6𝑥 − 6
So these equations represent linear functions.
and
2
𝑦= ,
𝑥
𝑥2 − 𝑦 = 0
and
You try!!
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the equation represent a linear or nonlinear function? Explain.
A linear function can be written in the form 𝑦 = 𝑚𝑥 + 𝑏.
𝑦 =𝑥+9
You can rewrite the
equation as 𝑦 = 1𝑥 + 9
Equation represents a
linear function.
𝑦 = 5 − 𝑥2
3𝑥
𝑦=
5
You can rewrite the
3
equation as 𝑦 = 𝑥 + 0
5
Equation represents a
linear function.
You cannot rewrite the
equation as 𝑦 = 𝑚𝑥 + 𝑏
Equation is a nonlinear
function.
So let’s review!
What have we covered so far?
1. Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
In a graph it looks like a ……
Line
In a table we look for what?
A constant rate of change in both x and y values.
In equations we need to be able to do what?
Write the equation in the form of 𝑦 = 𝑚𝑥 + 𝑏