Year 10 Interactive Maths - Second Edition

Gradient of a Straight Line

The gradient of a straight line is the rate at which the line rises (or falls) vertically for every unit across to the right. That is:

Note:

The gradient of a straight line is denoted by m where:

Example 3

Find the gradient of the straight line joining the points P(– 4, 5) and Q(4, 17).

Solution:

So, the gradient of the line PQ is 1.5.

Note:

If the gradient of a line is positive, then the line slopes upward as the value of x increases.

Example 4

Find the gradient of the straight line joining the points A(6, 0) and B(0, 3).

Solution:

Note:

If the gradient of a line is negative, then the line slopes downward as the value of x increases.

Applications of Gradients

Gradients are an important part of life. The roof of a house is built with a gradient to enable rain water to run down the roof. An aeroplane ascends at a particular gradient after take off, flies at a different gradient and descends at another gradient to safely land. Tennis courts, roads, football and cricket grounds are made with a gradient to assist drainage.

Example 5

A horse gallops for 20 minutes and covers a distance of 15 km, as shown in the diagram.

Find the gradient of the line and describe its meaning.

Solution:

In the above example, we notice that the gradient of the distance-time graph gives the speed (in kilometres per minute); and the distance covered by the horse can be represented by the equation:

The cost of transporting documents by courier is given by the line segment drawn in the diagram. Find the gradient of the line segment; and describe its meaning.

Solution:

So, the gradient of the line is 3. This means that the cost of transporting documents is $3 per km plus a fixed charge of $5, i.e. it costs $5 for the courier to arrive and $3 for every kilometre travelled to deliver the documents.

An alternative definition of the gradient is:

The gradient is the rate of change of one variable with respect to another.

Key Terms

gradient