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-values are increasing at a constant rate as the x-values increase. If the gradient is negative that means that the y-values are decreasing at a constant rate as the 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.

This image shows three increasing graphs at an increasing, constant or decreasing rate. Ask your teacher for more information.

An increasing function can increase at an increasing, constant or decreasing rate.

This image shows three decreasing graphs at an increasing, constant or decreasing rate. Ask your teacher for more information.

A decreasing function can decrease at an increasing, constant or decreasing rate.

Examples

Example 1

Christa went for a run. She started by gradually increasing her speed until she felt comfortable, she then kept her speed constant for a little while. Then she slowed down as she got tired towards the end of her run.

Which graph shows the total distance covered by Christa plotted against time?

This image shows the total distance covered plotted against time. Ask your teacher for more information.

Worked Solution

Create a strategy

Draw the curve representing an increasing rate then a constant rate and a decreasing rate.

Apply the idea

Christa continued on her run and did not turn back. So her distance was always increasing. This means the curve should always be increasing. So options A and F are not correct.

This image shows a curve increasing at an increasing rate in the first quadrant on the coordinate plane.

For the start of her run she was increasing her speed. This means the she was covering more distance at an increasing rate.

Increasing at an increasing rate will look like this on a graph.

So options B and C are incorrect.

In the next part of her run Christa is at a constant speed. So the next section of the curve should be an increasing straight line. So option D is incorrect.

In the last part of her run Christa is slowing down, so she is increasing at a decreasing rate which looks like the last part of this graph.

So option E is the correct answer.

Example 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.

2 graphs that shows the curve of rate of change of the height of water in a vase. Ask your teacher for more information.

Which vase could have produced the graph for Vase A?

This image shows a curved cylinder. Ask your teacher for more information.

This image shows a truncated cone with a wide base and a narrow top.

This image shows a truncated cone with a narrow base and a wide top.

Worked Solution

Create a strategy

Consider the rates of increase of the first graph.

Apply the idea

The narrower the vase the faster the height of the water increases. The wider the vase the slower the height of the water increases.

The graph shows that at first the height increased very fast and then slowed down. Since the water will fill the vase from the bottom to the top, the bottom of the vase needs to be narrower than the top.

So the answer is option D.

Which vase could have produced the graph for Vase B?

This image shows a curved solid where the base is wider than the top. Ask your teacher for more information.

This image shows a curved solid where the top is wider than the base. Ask your teacher for more information.

This image shows a curved solid where the top and base are wider than the middle. Ask your teacher for more information.

This image shows a curved solid where the top and base are narrower than the middle. Ask your teacher for more information.

Worked Solution

Create a strategy

Consider the rates of increase of the second graph.

Apply the idea

The graph shows that at first the height increased very fast and then slowed down and then sped up again. Since the water will fill the vase from the bottom to the top, the bottom and top of the vase need to be narrower than the middle.

So the answer is option D.

Example 3

Consider the function 2^x. Describe the rate of increase of the function.

As x increases, y increases at a decreasing rate.

As x increases, y increases at an increasing rate.

As x increases, y increases at a constant rate.

Worked Solution

Create a strategy

Make a table of values and look at how much the value of y changes by each time.

Apply the idea

Substituting each of the x-values in the table of values into the equation, 2^x, we have:

x	1	2	3	4
y	2	4	8	16

We can see that y increases by 1 then by 2 then by 4 then by 8.

So y is increasing at a faster and faster rate.

Option B is the correct answer.

Idea summary

The rate of change of a function is how quickly it increases or decreases.

If the y-values increase as the x-values increase it is an increasing function.

If the y-values decrease as the 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.

13.11 Rates of change

Ideas

Rates of change

Examples

Example 1

Example 2

Example 3

Outcomes

MA5.3-4NA

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