Gradients & Rates of Change (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Gradient of a Straight Line

How do I find the gradient of a straight line?

• The gradient of a straight line is given by

  • or

    • Where two co-ordinates on the line are given by and

  • You may also see this phrased as "rise over run"

• In the straight line equation,

  • is the gradient of the line

• If the equation of the line is written in a different format, rearrange into the form to find the gradient

Interpreting the gradient as a rate of change

• The gradient is the rate of change of the line as increases

• When modelling, the variables will be real life quantities rather than and

• For example, when modelling:

  • The volume of water in a cup as it is filled

    • Volume would be on the -axis, and time would be on the -axis

    • The gradient would be volume per unit of time

    • E.g. ml per second

  • Rate of pay per hour

    • Pay would be on the -axis, and time would be on the -axis

    • The gradient would be pay per unit of time

    • E.g. £ per hour

  • The extension of a spring as a force is applied

    • Extension of the spring would be on the -axis and the applied force would be on the -axis

    • The gradient would be the extension per unit of force

    • E.g. cm per N (centimetres per Newton)

Gradient at a Point on a Curve

How do I find the gradient of a curve at a point?

• The gradient of a graph at any point is equal to the gradient of the tangent to the curve at that point

• Remember that a tangent is a line that just touches a curve (and doesn’t cross it)

• To find an estimate for the gradient:

  • Draw a tangent to the curve

    • This will only be done by eye, so the gradient will be an estimate

  • Find the gradient of the tangent by finding or "rise over run"

• For the above diagram, the gradient would be

  • Always check if the gradient should be positive or negative

    • An upward slope has a positive gradient

    • A downward slope has a negative gradient

How do I interpret the gradient of a graph at a point?

• As discussed above for linear graphs, when modelling a real-life scenario, consider the meaning and units of the and axes

  • The gradient will be the -axis units, divided by the -axis units

    • E.g. millilitres per minute, or kilometres per litre

• On a curve, the gradient is constantly changing

  • It is important to understand that the gradient at a point is the gradient at that point only

    • Not the gradient for the whole graph

• If the gradient is zero at a point on the graph

  • The tangent at that point will be horizontal

  • This point is either a local minimum or a local maximum

  • They are described as "local" because whilst they are a peak or a trough, they may not be the maximum or minimum point of the graph overall

    • An example of this is shown in the below image

How do I find an average rate of change?

• An average rate of change can be found by:

  • Selecting a point on the curve at the start

  • Selecting a point on the curve at the end

  • Drawing a line between them (a chord)

  • Finding the gradient of the chord joining the two points

• An example of this method is shown in part (b) of the worked example below