Gradients (AQA GCSE Further Maths): Revision Note

Gradients of lines

What is the gradient of a line?

• The gradient is a measure of how steep a 2D line is

  • A large value for the gradient means the line is steeper than for a small value of the gradient

    • 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 from left to right - "uphill"

  • A negative gradient means the line goes downwards from left to right - "downhill"

• In the equation for a straight line, $y=mx+c$, the gradient is represented by $m$

  • The gradient of $y=-3x+2$ is −3

How do I find the gradient of a line?

• The gradient can be calculated using

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

• You may see this written as $\frac{\mathrm{rise}}{\mathrm{run}}$ instead

  • $\frac{\mathrm{d}y}{\mathrm{d}x}$ may even be used which links to the work on Calculus

    • this can be read as "the difference in y divided by the difference in x"

• You need to know two coordinates a line passes through to find its gradient

  • If given two coordinates $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ the gradient of the line joining them is

$\frac{y_2-y_1}{x_2-x_1}\mathrm{or}\frac{y_1-y_2}{x_1-x_2}$

• The order of the coordinates must be consistent on the numerator and denominator

  • i.e. ("Point 2" – "Point 1") or ("Point 1" – "Point 2") for both

  • If given a diagram of a straight line you will need to pick two points the line passes through

    • If possible, pick whole number coordinates

      • positive numbers are easier to work with than negatives!

      • try not to pick coordinates that are close together

Parallel & perpendicular gradients

What are parallel lines?

• Parallel lines are equidistant meaning they never meet

• Parallel lines have equal gradients

What are perpendicular lines?

• Perpendicular lines meet at right angles

• The product of their gradients is -1

How do I tell if lines are parallel or perpendicular?

• Rearrange equations into the form y = mx + c

  • m is the gradient