Gradient of a Line (SQA National 5 Maths): Revision Note

Determining the gradient of a straight line between two points

What is the gradient of a line?

• The gradient is a measure of how steep a straight line is

• A gradient of 3 means:

  • For every 1 unit to the right, go up by 3

• A gradient of -4 means:

  • For every 1 unit to the right, go down by 4 

• A gradient of 3 is steeper than a gradient of 2

  • A gradient of -5 is steeper than a gradient of -4

• A positive gradient means the line goes upwards (uphill)

  • Bottom left to top right 

• A negative gradient means the line goes downwards (downhill)

  • Top left to bottom right

How do I find the gradient of a line?

• Find two points on the line and draw a right-angled triangle

  • Then $\mathrm{gradient}=\frac{\mathrm{change}\mathrm{in}y}{\mathrm{change}\mathrm{in}x}$

  • Or, in short, $\frac{\mathrm{rise}}{\mathrm{run}}$ 

    • The rise is the vertical length of the triangle

    • The run is the horizontal length of the triangle

  • Put the correct sign on your answer

    • Positive for uphill lines

    • Negative for downhill lines

  • You can also find gradient of a line between two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ 

    • Use the formula $\mathrm{gradient}=\frac{y_2-y_1}{x_2-x_1}$

How do I draw a line with a given gradient?

• To draw the gradient $\frac{2}{3}$

  • The rise is 2

  • The run is 3

  • It is positive (uphill)

    • Move 3 units to the right and 2 units up

• To draw the gradient $-5$ make it a fraction, $-\frac{5}{1}$

  • The rise is 5

  • The run is 1

  • It is negative (downhill)

    • Move 1 unit to the right and 5 units down