The rate of change is how one variable changes with respect to another. If the change is constant, there is a linear relationship between the variables.

The gradient is a measure of how steep a line is. It is calculated by finding the rate of change in the $$y$$-direction with respect to the positive $$x$$-direction.

Calculating the rate of change

Maths

The work on gradient in the prior unit is expanded further in this unit by considering gradient as the rate of change and interpreting this in context. The gradient of the tangent is introduced as a way of estimating the gradient of a curve.

Real-life graphs

Every one New Zealand Dollar is worth two British Pounds.

Every one British Pound is worth two New Zealand Dollars.

For every change of $$+1$$ in $$£$$ there is a change of $$+2$$ in $${$}$$

For every change of $$+1$$ in $${$}$$ there is a change of $$+2$$ in $$£$$

For every change of $$+10$$ in $${$}$$ there is a change of $$+20$$ in $$£$$

This is a conversion graph for British Pounds ($$£$$) to New Zealand Dollars ($${$}$$) and the gradient of the line is $$2$$. Which of the below statements are accurate?

What is the speed of the vehicle modelled in this distance-time graph?

The rate of change of the cost is $$£2$$ per mile.

Every extra mile travelled adds $$£2$$ to the cost.

This graph shows the charges of a taxi firm in $$£$$ for every mile ($$m$$) travelled. Which of these statements are accurate about the rate of change for this taxi firm's charges?

How much faster is this vehicle travelling in the first hour of its journey versus the sixth hour?

In this table there is a __________ rate of change in the $$y$$ values.

In this table for every change of $$+1$$ in $$x$$ there is a change of _________ in $$y$$.

The graph does not intercept the $$y$$-axis at the origin.

How do we know there is not a constant rate of change between the $$x$$ and $$y$$ variables in this graph?

When we find the gradient on a distance-time graph we have calculated the __________ of the particle being modelled.

Starter Quiz