We can identify linear and non-linear relations by looking at their graphs, tables of values, or equations.
Graphs
The graphs of linear relations will have a constant rate of change, so will be a straight line. The slope can be positive, negative or zero.
Linear relations
The graphs of a non-linear relation will be a curve. For example:
Non-linear (quadratic) relation.
Practice question
Question 1
Consider the graph shown below.
Does this graph represent a linear function?
YesANoBWhich of the following explanations are accurate?
Select all that apply.
The relationship is linear because it grows towards infinity in one direction.
AThe relationship is linear because the graph is a straight line.
BThe relationship is linear because it has a constant slope.
CThe relationship is linear because it passes through the origin.
D
Tables of values
We can calculate the first differences to determine if the relation in a table of values is linear or not. The first differences are the difference between consecutive $y$y-values. We need to check that the $x$x-values are also changing by the same amount between each pair of consecutive rows.
Worked example
EXAMPLE 1
Edna is selling cookies to raise money for charity. The table below shows the cumulative number of cookies she has sold each hour for the first three hours.
| Time (hours) | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
|---|---|---|---|---|---|
| Total cookies sold | $22$22 | $44$44 | $66$66 | $88$88 | $110$110 |
Determine whether or not Edna's cookie sales represent a linear relation.
Think: Notice that the $x$x-values (time) is going up by $1$1 each time. We just need to check that the $y$y-values (cookies) are also going up by a constant each time to see if it is linear.
Do: We need to subtract the previous y-value for each column. We can see that each column has a first difference of $22$22, so this is a constant rate of change, so it is a linear relation.
Calculating the first differences
Linear relations (where the equations have a highest power of $1:\quad y=3x-6$1: y=3x−6) have the same first difference between the $y$y-values as the $x$x-values increase by $1$1.
Quadratic relations (where the equations have a highest power of $2:\quad y=x^2$2: y=x2) have the same second difference between the $y$y-values as the $x$x-values increase by $1$1.
Cubic relations (where the equations have a highest power of $3:\quad y=x^3$3: y=x3) have the same third difference between the $y$y-values as the $x$x-values increase by $1$1.
example 2
Vincent is training for a remote control plane aerobatics competition. He wants to fly his plane through the following points, where $y$y is the height in metres of the plane from the ground, and $x$x is the horizontal distance in metres of the plane from its starting point.
| Horizontal distance, $x$x | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
|---|---|---|---|---|---|
| Height, $y$y | $3$3 | $12$12 | $27$27 | $48$48 | $75$75 |
Determine whether Vincent wants to fly his plane in a straight line or parabolic (quadratic) curve or neither.
Think: Notice that the $x$x-values (time) is going up by $1$1 each time. We just need to check the differences between the $y$y-values. If the first differences are the same, then the relation is linear so the plane is moving in a straight line. Otherwise, if the second differences are the same, then the relation is quadratic so the plane is moving in a parabolic shape.
Do: We need to subtract the previous y-value for each column:
| $x$x | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
|---|---|---|---|---|---|
| $y$y | $3$3 | $12$12 | $27$27 | $48$48 | $75$75 |
| 1st difference | $-$− | $(12-3=)\quad9$(12−3=) 9 | $15$15 | $21$21 | $27$27 |
Since the first differences are not the same, we know that the relation is not linear, so Vincent does not want to fly his plane in a straight line.
Think: Now we need to check the second differences to see if the relation is quadratic.
| $x$x | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
|---|---|---|---|---|---|
| $y$y | $3$3 | $12$12 | $27$27 | $48$48 | $75$75 |
| 1st difference | $-$− | $9$9 | $15$15 | $21$21 | $27$27 |
| 2nd difference | $-$− | $-$− | $(15-9=)\quad6$(15−9=) 6 | $6$6 | $6$6 |
We can see that the second differences are the same so the relation is quadratic. So Vincent wants to fly his plane in a parabolic shape.
Practice question
Question 2
Consider the table of values below.
| Time period | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
|---|---|---|---|---|---|
| Revenue | $-4$−4 | $3$3 | $10$10 | $17$17 | $24$24 |
Is revenue changing at a constant rate?
Yes
ANo
BIs the relationship between time and revenue linear?
Yes
ANoB
Equations
A linear equation will have a variable both variables with degree $1$1 and can involve coefficients and constants. Often we write a linear equation in form $y=ax+b$y=ax+b.
Examples of linear relations are:
- $y=3x+5$y=3x+5
- $y=3$y=3
- $y=5-\frac{x}{2}$y=5−x2
-
$2x-3y=6$2x−3y=6
Non-linear relation equations do not have a degree (highest power) of $1$1, and can be in completely different forms.
Examples of non-linear relations are:
- $y=3x^2+5$y=3x2+5
- $y=\frac{3}{x}$y=3x
- $y^2=5-\frac{x}{2}$y2=5−x2
-
$y=4\times8^x$y=4×8x
Practice questions
Question 3
Consider the relation $y=3^x$y=3x.
Does this relation represent a linear function?
YesANoBWhich of the following explanations are accurate?
Select all that apply.
The function is not linear because it is increasing.
AThe function is not linear because if we sketched the function, its graph would be a straight line.
BThe function is not linear because each time $x$x increases by $1$1, the $y$y-value does not increase by a constant amount.
CThe function is not linear because it cannot be written in the form $y=ax+b$y=ax+b.
D
question 4
Which of the following functions are linear?
Select all that apply.
$y=-5\left(x-4\right)+3$y=−5(x−4)+3
A$y=9\times5^x$y=9×5x
B$y=-2\times\left(\frac{1}{3}\right)^x$y=−2×(13)x
C$y=3x+7$y=3x+7
D