The gradient

The gradient tells us how steep a line is, therefore the bigger the gradient the steeper the line is.

A positive gradient is a straight line which slopes up to the right.

A negative gradient is a straight line which slopes down to the right.

Gradients of special types of lines

Parallel lines have the same gradient

2 parallel diagonal lines

Vertical lines have a gradient which is undefined

Equation x = a

Vertical line

Horizontal lines have a gradient of zero

Equation y = b

Horizontal line

One of the formulae used to find the gradient of a straight line is:

Gradient\,of\,a\,slope = \frac{{vertical\,distance}}{{horizontal\,distance}}

Right angle triangle showing vertical and horizontal dimensions, with the gradient slope highlighted

Now try some example questions.

Question

The community centre is getting a new ramp onto their side entrance.

Calculate the gradient of the ramp using the diagram below.

Diagram of a 26cm x 78cm right-angled triangle

The gradient can be worked out as a simplified fraction or a decimal fraction. Gradient is positive since the line slopes up to the right.

Gradient = \frac{{vertical\,distance}}{{horizontal\,distance}}

= \frac{{26}}{{78}}

=\frac{13}{39}

=\frac{1}{3}

or

= 26 \div 78

= 0.333...

= 0.3\,(to\,1\,d.p.)

Question

Calculate the gradient of the line shown below.

Gradient line on graph using the grid to plot a right angled triangle

Gradient is negative since the line slopes down to the right.

Gradient\, = \frac{{vertical\,distance}}{{horizontal\,distance}}

-\frac{5}{2}

Question

Calculate the gradient of the slope below.

Diagram of a 5cm x 20cm right-angled triangle

Gradient is negative since the line slopes down to the right.

Gradient\, = \frac{{vertical\,distance}}{{horizontal\,distance}}

= \frac{{ - 5}}{{20}}

=  - 5 \div 20

=  - 0.25\,(to\,2\,d.p.)

Question

Calculate the gradient of the slope below.

Diagram of a 10m x 45m right-angled triangle

Gradient is positive since the line slopes up to the right.

Gradient\, = \frac{{vertical\,distance}}{{horizontal\,distance}}

= \frac{{10}}{{45}}

= \frac{2}{9}

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