Algebra - Graphs: plot and interpret graphs (including reciprocal graphs) and graphs of non-standard functions in real contexts to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration (Foundation tier)

The idea

Real-context graphs show relationships between quantities. Always check axis labels and units. For kinematics, distance-time graphs show a journey; the gradient (steepness) represents speed, and a horizontal line means stationary. Speed-time graphs show how speed changes; the gradient represents acceleration (how quickly speed is changing), and a horizontal line means constant speed. To find approximate solutions, draw lines from a known value on one axis to the curve, then to the other axis. Non-standard graphs model unique situations, like a container filling. Reciprocal graphs (y = k/x) show that as one quantity increases, the other decreases, like speed and time for a fixed distance.

What earns the marks

• Plotting points accurately from a table of values.
• Joining the points with a suitable line or curve based on the context.
• Reading a value correctly from a graph, with clear construction lines shown.
• Correctly interpreting a feature of a distance-time graph, e.g., identifying the steepest section as the highest speed.
• Calculating acceleration from the gradient of a section on a speed-time graph.
• Describing the relationship shown by the graph in the context of the problem, using correct units.

Common mistakes and how to put them right

• Confusing a horizontal line on a distance-time graph (which means stationary) with a horizontal line on a speed-time graph (which means constant speed).

  Put it right: Check the vertical axis label. If it's distance, no change means no movement. If it's speed, no change means speed is steady.

• Misreading the scale on an axis, particularly when each square represents more than one unit.

  Put it right: Before reading any values, determine what one small square is worth on each axis.

• Calculating speed from a speed-time graph instead of acceleration from its gradient.

  Put it right: The gradient of a speed-time graph is acceleration (change in speed ÷ time). The speed at any instant is read directly from the vertical axis.

Exam tips

• Always draw construction lines (e.g., from an axis value to the curve and then to the other axis) on the graph. They show your method and can earn marks even if your final reading is slightly out.
• When asked to interpret a graph, write a sentence that uses the words from the axis labels to describe what is happening in the real-world context, e.g., 'The car is accelerating between 10 and 20 seconds'.