Rates-Of-Change-Of-Graphs Wjec-Eduqas Higher

Rates of change of graphs

What is a rate-of-change graph?

• A rate-of-change graph usually shows how a variable changes with time

• The following are examples of rates-of-change graphs:

  • Speed against time

    • Speed is the rate of change of distance as time increases

  • Acceleration against time

    • Acceleration is the rate of change of velocity as time increases

  • The depth of water against time (e.g. in a container as it is filled with water)

Can rates-of-change graphs not be against time?

• More generally, rate-of-change graphs can show any two different variables plotted against each other, not just time

  • E.g. the volume of air inside an inflating balloon plotted against the balloon’s radius

    • This shows the rate of change of volume as radius increases

  • E.g. the number of ice-creams sold plotted against the weather temperature

    • This shows the rate of change of number of ice-creams as temperature increases

How can I use gradients to find rates of change?

• The gradient of the graph of y against x represents:

  • the amount of change in y for every 1 unit of increase in the x-direction

    • This is the amount of y per unit of x

  • This is called the rate of change of y against x

• The units of gradients are the units of the y-axis, divided by the units of the x-axis

  • E.g. If the graph shows volume in cm³ on the y-axis and time in seconds on the x-axis, the rate of change is measured in cm³/s (or cm³s^-1)

• If the graph is a straight line the rate of change is constant

  • If the graph is horizontal, the rate of change is zero

    • y is not changing as x changes

How can I use tangents to find rates of change?

• If the graph is a curve, you can draw a tangent at a point on the graph and find its gradient

  • This will be an estimate of the rate of change of y against x

• The rate of change is greater when the graph is steeper

  • In the below image

    • tangents drawn at points A and B show the graph is steeper at B

    • therefore the rate of change at B is greater

• On a distance-time graph, a tangent at a point on the curve can be used to estimate the velocity at that particular time

• On a speed-time graph, a tangent at a point on the curve can be used to estimate the acceleration at that particular time