Rates-of-Change Graphs (Cambridge International IGCSE Mathematics)

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)

Do rates-of-change graphs always include time as a variable?

• 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 speed of an object plotted against its distance from the starting point

    • This shows the rate of change of the speed of an object as it moves further away from the starting point

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

Exam Hint The units of the gradient can help you understand what is happening in the context of an exam question.

For example, if the y-axis is in dollars and the x-axis is in hours, the gradient represents the change in dollars per hour.

Worked Example (a) Each of the graphs below shows the depth of water, d cm, in different containers that are being filled from a running tap of water with a constant flow rate.

Match each of the graphs 1, 2, 3, 4 with the containers A, B, C, D.

Answer:

Considering graph 1: the gradient is constant This means the rate of change is constant So the depth increases at the same rate throughout

This matches container D which has vertical sides, so depth increases uniformly

Final answer: Graph 1 is container D

Considering graph 2: the gradient starts shallow and becomes steeper, meaning that the depth increases faster and faster at the end

This matches container A, which gets narrower towards the top, causing the depth to increase faster at the end

Final answer: Graph 2 is container A

Considering graph 3: the gradient starts steep and becomes shallower, meaning that the depth increases at a slower and slower rate as time increases

This matches container B, which gets wider towards the top, causing the depth to increase more slowly at the end

Final answer: Graph 3 is container B

Considering graph 4: the gradient starts steep, then becomes shallow, then becomes steep again. This means that the depth increases quickly, then slowly, then quickly again

This matches container C, which is narrow at the bottom (fast depth increase), gets wider in the middle (slower depth increase) and narrow again at the top (faster depth increase)

Final answer: Graph 4 is container C

(b) The graph below shows a model of the volume, v litres, of diesel fuel in the tank of George’s truck after it has travelled a distance of d kilometres.

(i) Find the gradient of the graph, stating its units clearly.

Answer:

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

$\mathrm{Gradient}=\frac{-54\mathrm{litres}}{600\mathrm{kilometres}}$

Final answer: -0.09 litres per kilometre

(ii) Interpret what the gradient of this graph represents.

Answer:

Consider the units of the gradient: litres per kilometre An increase in 1 kilometres causes a decrease of 0.09 litres in diesel fuel

Final answer: The gradient represents the amount of diesel fuel used to travel one kilometre Travelling 1 km uses up 0.09 litres of diesel fuel

(iii) Give one reason why this model may not be realistic.

Answer:

Final answer: Fuel may not be consumed at a constant rate