GCSE · Maths · AQA · Spec 8300 · Foundation

Real-context graphs and kinematic problems

One line on a grid can tell you when someone left, where they stopped, which bit they rushed and when they got home — once you can read it.

Maya's bike ride, told by one line

time since leaving home (minutes): 25. distance from home (km): 5

Slide the point along the line. For each piece, ask: moving away, stopped, or heading home? Faster or slower than before?

Axes before shape

Same flat line, two different stories

Two graphs each have a flat, horizontal piece from 10 to 20 seconds. Graph A has 'distance (m)' up the side. Graph B has 'speed (m/s)' up the side.

What is the object doing during the flat piece on each graph?

Both show it stopped. Flat means not moving.A is stopped. B is moving at a steady speed.Both show it moving at a constant speed.Both show it moving along flat ground instead of up a hill.

How sure are you?

I'm sureI think soI'm guessing

Both show it stopped. Flat means not moving.

This is a common misconception

That's true for Graph A only. On a distance-time graph, flat means the distance isn't changing, so the object is standing still. But Graph B measures speed. Flat there means the speed isn't changing — it could be cruising along at 8 m/s the whole time. Always check what's up the side before you trust the shape.

A is stopped. B is moving at a steady speed.

This matches the evidence

Exactly. You read the axis first. Flat on distance-time: the distance stays the same, so it's stationary. Flat on speed-time: the speed stays the same, so it's moving at a constant speed. On a speed-time graph, a line sloping up means it's speeding up (accelerating), and a line sloping down means it's slowing down (decelerating).

Both show it moving at a constant speed.

This is a common misconception

Right for Graph B, not for Graph A. On a distance-time graph, a steady speed shows up as a straight SLOPING line — the distance grows by the same amount every second. A flat line means the distance isn't growing at all, so the object has stopped.

Both show it moving along flat ground instead of up a hill.

This is a common misconception

This is a sneaky one: treating the graph as a picture of the route. It isn't a map or a side view of a hill. The height of the line is a distance (Graph A) or a speed (Graph B), so an upward line means 'further away' or 'faster' — never 'uphill'.

Read what's up the side before you read the shape. Distance-time: flat = stopped. Speed-time: flat = steady speed, up = speeding up, down = slowing down.

Which shape?

Sort each situation by the shape of its graph

Which shape of graph does each situation make?

Still to sort

Straight line (0)

Equal steps across always change the other quantity by the same amount.

Where the line is: A straight line ADDS (or takes away) the same amount each step. An exponential curve MULTIPLIES by the same amount each step.

Reciprocal curve (0)

The two quantities multiply to the same total every time: double one and the other halves.

Where the line is: A reciprocal curve never touches either axis. An exponential decay curve starts at a value on the vertical axis and only gets close to the horizontal one.

Exponential curve (Higher only) (0)

Higher only: each step multiplies by the same number, so growth gets steeper and steeper and decay levels off towards zero.

Where the line is: Doubling is not the same as adding: going 1, 2, 4, 8 gets steeper and steeper, while going 1, 2, 3, 4 climbs at the same rate.

6 of 6 still to sort.

Ask what happens each time one quantity goes up by the same step.

Plot it, then let the graph solve it

Problem

A charity walk is 12 km long. Draw the graph of the time taken, t hours, against the average walking speed, v km/h, for speeds from 2 to 6 km/h. Use it to find the average speed needed to finish in 2.5 hours.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Read the axes first, then read the line — and let the graph answer the question for you.

What you need to know

Read the axis labels and units before you look at the shape: they tell you what the height of the line means.
On a distance-time graph, a rising line means moving away, a flat line means stationary, a falling line means coming back, and a steeper line means faster.
On a speed-time graph, a flat line means a constant speed, a rising line means speeding up (acceleration) and a falling line means slowing down (deceleration).
When two quantities multiply to a fixed total, their graph is a reciprocal curve: double one and the other halves, and the curve never touches either axis.
Higher only: when each step multiplies by the same number, the graph is an exponential curve — growth gets steeper and steeper, decay levels off towards zero.
To solve a problem graphically, go from the known value on its axis to the curve, then across or down to the other axis. The answer is an estimate: give units and say what it means.

The big picture

A real-life graph tells a story, and you read it by checking the axes before the shape. On a distance-time graph, rising means moving away, flat means stopped, falling means coming back and steeper means faster; on a speed-time graph, flat means a steady speed, rising means speeding up and falling means slowing down. Two quantities that multiply to a fixed total make a reciprocal curve, and (Higher) multiplying by the same number each step makes an exponential curve. To solve a problem graphically, go from the known value to the curve and read the other axis, giving an approximate answer with units.

Key points

1A graph is not a picture of the route: an upward line on a distance-time graph means further from the start, not uphill.

2The same flat line tells two stories: stopped on a distance-time graph, steady speed on a speed-time graph.

3Steepness is a rate: the steeper a distance-time line, the faster the movement; the steeper a speed-time line, the faster the speed is changing.

4A horizontal line drawn across a graph can meet it more than once, so a question like 'when was she 3.5 km away?' can have two answers.

5Plot reciprocal and exponential points, then join them with a smooth curve — straight segments add corners that aren't really there.

6Straight line: add the same amount each step. Reciprocal: the product stays the same. Exponential (Higher): multiply by the same number each step.

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Worked example

Problem

A car's speed-time graph is made of straight lines joining (0, 0), (10, 20), (30, 20) and (40, 0), with time in seconds across and speed in m/s up. Describe the car's motion, and estimate the times when it was travelling at 10 m/s.

⚠ Watch out

Reading the shape before the axes — for example, saying a flat line on a speed-time graph means 'stopped'. It means the speed isn't changing, so the object is moving at a steady speed.

🧠

Memory hook

Axes first, then the story. Ask 'what's up the side?' before you ask 'what's the line doing?' — because the same flat line can mean 'stopped' or 'cruising'.

✓

Check yourself

A distance-time graph joins (0, 0), (30, 3) and (50, 3), in minutes and km. What is happening from 30 to 50 minutes? (Answer: stopped for 20 minutes, 3 km from the start.)

Flashcards

(10)

What should you read first on any real-life graph?

The axis labels and units — what's across and what's up. They tell you what the height of the line means.

Distance-time graph: what do rising, flat and falling pieces mean?

Rising: moving away from the start. Flat: stationary. Falling: coming back towards the start.

Distance-time graph: two rising pieces, one steeper. What does the steeper one tell you?

The object is moving faster during the steeper piece.

Speed-time graph: what do flat, rising and falling pieces mean?

Flat: constant speed. Rising: speeding up (accelerating). Falling: slowing down (decelerating).

Why is a graph not a picture of the route?

The height of the line is a quantity like distance from home or speed, so an upward line means 'further away' or 'faster', never 'uphill'.

What makes a real-life graph a reciprocal curve?

The two quantities multiply to a fixed total — like speed × time for a fixed distance. Double one and the other halves; the curve never touches either axis.

How should plotted points on a curved graph be joined?

With a smooth curve through every point — not straight segments from point to point.

How do you use a graph to solve a problem?

Start at the known value on its axis, go across (or up) to the curve, then read the other axis. Give an approximate answer with units, in context.

Why can a question like 'when was she 3.5 km from home?' have two answers?

A horizontal line at 3.5 can meet the graph twice — once on the way out and once on the way back.

Higher: what makes an exponential graph, and how does it look?

Each step multiplies by the same number. Growth gets steeper and steeper; decay levels off towards zero.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Maya's bike ride, told by one line

Same flat line, two different stories

Sort each situation by the shape of its graph

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