Gradients (AQA GCSE Further Maths)

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, , the gradient is represented by 

  • The gradient of is −3

How do I find the gradient of a line?

• The gradient can be calculated using

• You may see this written as  instead

  •  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  and  the gradient of the line joining them is

• 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