0580

Real Life Graphs

Coordinate Geometry and Graphs

When the rate itself changes

  • On a curve no two points have the same steepness, so the rate of change differs everywhere and one gradient cannot describe it
  • To find the rate at one moment, draw a tangent at that point and work out its gradient, exactly as in the further graphs topic
  • On a travel graph a tangent gives the speed at that instant; on a speed–time graph it gives the acceleration
  • A curve getting steeper means the rate is increasing, and a curve flattening out means it is decreasing

Matching a graph to a situation

  • Questions often show a container being filled at a steady rate and ask which depth-against-time graph fits it
  • Reason from the width of the container, since a steady inflow raises the level faster where the container is narrower
    • Straight sides give a straight line, because the depth rises at one steady rate
    • A container narrowing towards the top gives a curve that steepens
    • A container widening towards the top gives a curve that flattens
  • The same reasoning matches any pair of quantities, so read the axis labels and ask what one unit of the horizontal quantity does to the vertical one
Exam tip

A tangent must be a ruled straight line touching the curve, not a chord joining two points on it, and it must not leave a gap where it meets the curve. Mark schemes award a mark for the tangent itself before any gradient is read from it.

Worked example

Matching each container to its graph

Each graph shows the depth of water, d cm, in one of the containers as it is filled from a tap running at a constant rate. Match each graph to its container.

Four containers labelled A to D above four depth-against-time graphs numbered 1 to 4. A is a triangular prism standing on its base so it narrows towards the top, B is a container with sloping sides that widens towards the top, C is a round flask that is narrow at the bottom, wide in the middle and narrow again at the neck, and D is a cuboid with vertical sides. Graph 1 is a straight line, graph 2 curves upwards getting steeper, graph 3 rises steeply then flattens, and graph 4 is steep, then shallow, then steep again
Source: Rates of change of graphs by Save My Exams

Solution:

  • The water goes in at a steady rate, so the depth rises fastest where the container is narrowest
  • Graph 1 is straight, so the depth rises at one steady rate all the way up, which needs a container of constant width
  • Graph 1 is container D, the cuboid with vertical sides
  • Graph 2 starts shallow and steepens, so the depth rises faster and faster, which needs a container narrowing towards the top
  • Graph 2 is container A
  • Graph 3 starts steep and flattens, so the depth rises more and more slowly, which needs a container widening towards the top
  • Graph 3 is container B
  • Graph 4 is steep, then shallow, then steep again, which matches narrow, then wide, then narrow
  • Graph 4 is container C, the round flask
  • Read the shape of each curve first and describe it in words before looking at the containers, because the reasoning is the same every time