Linear functions have a constant gradient. This can be thought of as the rate of change of the function. If the gradient is positive, that means that the $y$y-values are increasing at a constant rate as the $x$x-values increase. If the gradient is negative that means that the $y$y-values are decreasing at a constant rate as the $x$x-values increase. These are examples of increasing functions and decreasing functions respectively.
Non-linear functions also have a rate of change. Unlike linear functions, the rate of change is variable. That is, the rate of change can increase or decrease.
An increasing function can increase at an increasing, constant or decreasing rate
An decreasing function can increase at an increasing, constant or decreasing rate
The rate of change of a function is how quickly it increases or decreases.
If the $y$y-values increase as the $x$x-values increase it is an increasing function.
If the $y$y-values decrease as the $x$x-values increase it is a decreasing function.
The rate of change of a linear function is constant while the rate of change of a non-linear function is variable.
Practice questions
Question 1
Christa went for a run. She started by gradually increasing her speed until she felt comfortable, then kept her speed constant for a little while. She then slowed down as she got tired towards the end of her run.
Which graph shows the total distance covered by Christa plotted against time?
A
B
C
D
E
F
Question 2
Ned has two graphs. Each graph shows how the height of the water in a vase changes as a constant flow of water is poured in.
 |
 |
| Vase $A$A | Vase $B$B |
Which vase could have produced the graph for vase $A$A?
A
B
C
DWhich vase could have produced the graph for vase $B$B?
A
B
C
D
Question 3
Consider the function $y=2^x$y=2x.
Describe the rate of increase of the function.
As $x$x increases, $y$y increases at a decreasing rate.
AAs $x$x increases, $y$y increases at an increasing rate.
BAs $x$x increases, $y$y increases at a constant rate.
C