Gradient is a measure of how steep a slope or a line is. Gradients can be calculated by dividing the vertical height by the horizontal distance.

Before doing this section you should look at the National 4 Lifeskills Maths section on Gradient of a slope.

The National 4 section shows how to calculate the gradient of a slope using the vertical height and horizontal distance.

A similar approach can be used for calculating the gradient of a line between two known points on a coordinate diagram.

Calculate the gradient of the line joining point \(A\,\,(3,2)\) and the point \(B\,\,(11,6)\).

Plot the points on square paper and you will see that line \(AB\) is sloping up, therefore the gradient is positive.

The vertical height can be found by subtracting the \(y-coordinate\)of \(A\) from the \(y-coordinate\)of \(B\).

\[=6-2=4\]

The horizontal distance can be found by subtracting the \(x-coordinate\)of \(A\) from the \(x-coordinate\)of \(B\).

\[=11-3=8\]

Make a right-angled triangle with the line \(AB\) as hypotenuse.

Gradient of line \(AB\, = \frac{{vertical\,height}}{{horizontal\,distance}}\)

\[ =\frac{4}{8} \]

\[ =\frac{1}{2} \]

Now try this question.

- Question
Calculate the gradient of the line joining point \(A\,\,(-2,8)\) and the point \(B\,\,(5,1)\).

Plot the points on square paper and you will see that line \(AB\) is sloping down, therefore the gradient is negative.

Make a right angled triangle with the line \(AB\) as hypotenuse.

Gradient of line \(AB\, = \frac{{vertical\,height}}{{horizontal\,distance}}\)

\[ \frac{7}{7}=1 \]

Gradient of line \(AB = -1\) (negative one)