GCSE · Maths · AQA · Spec 8300 · Higher

Gradients and area under graphs (Higher)

A speedometer shows your speed at one instant. But a curved distance–time graph is never the same steepness twice, so how do you read a speed off it?

Higher

Squeeze a chord into a tangent

Second point at 0 s

Second point at: 0 s. Gradient of red line: 6.0 m/s

Red line. While its two ends sit on the curve it is a chord, and its gradient is the average speed between those two times. Start at 0 s: 36 m in 6 s is 6 m/s on average. Slide the second point towards t = 6 s and the gradient climbs: 9 m/s from t = 3 s, 11 m/s from t = 5 s. Close the gap completely and the line only touches the curve at t = 6 s. That is the tangent, and its gradient, 12 m/s, is the speed at that instant.

A cyclist speeding up from rest: distance in metres against time in seconds. The big dot marks the moment we care about, t = 6 s. The red line joins it to a second point on the curve.

Higher

Same shape, different story

Distance–time graphvsSpeed–time graph

Start with the flat line: it's the one that catches people out.

Focus

A flat section

Distance–time graph

No distance is being covered. The object isn't moving.

Speed–time graph

The speed is steady, not zero: the object keeps moving at that speed.

The insight

Same flat line, opposite stories. On a speed–time graph, flat only means stopped if the line sits on the time axis itself, at 0 m/s.

What the gradient gives

Distance–time graph

The speed: distance divided by time, for example metres per second.

Speed–time graph

The acceleration: how fast the speed is changing.

Sloping down

Distance–time graph

The gradient is negative, because distance gets smaller as time gets larger.

Speed–time graph

The gradient is negative, which means deceleration: the object is slowing down.

The area between the graph and the time axis

Distance–time graph

Not needed: you read distance straight off the vertical axis.

Speed–time graph

The distance travelled, because speed × time = distance.

Higher

Your turn to supply the steps

Estimate a distance from a curved speed–time graph

A car's speed–time graph is a curve. On the grid, 1 square across is 5 s and 1 square up is 2 m/s. At t = 0, 10, 20 and 30 s the curve is 2, 5, 6.5 and 7 squares above the time axis. Use three trapezia to estimate the distance the car travels in the first 30 s.

Split the area from t = 0 to t = 30 s into three strips, each 10 s wide.Each strip has straight vertical sides and a slanted top that follows the curve closely: a trapezium.
missing step

Which line is step 2?

Use the heights in squares as they are: 2, 5, 6.5 and 7.Turn each height into a speed with the vertical scale: 2, 5, 6.5 and 7 squares × 2 m/s = 4, 10, 13 and 14 m/s.Turn each height into a speed with the time scale: 2, 5, 6.5 and 7 squares × 5 = 10, 25, 32.5 and 35 m/s.

Higher

Which do you think?

The £2-a-mile taxi

A taxi fare graph is a straight line. It crosses the fare axis at £5, before the taxi has moved at all, and rises £2 for each extra mile.

What does a 5-mile journey cost, and what does the gradient of £2 per mile mean?

It's £2 a mile, so 5 miles costs 5 × £2 = £10.5 miles costs £15, and the gradient means each extra mile adds £2 to the fare.The gradient is the cost per mile overall, so the fare divided by the miles is always £2.

How sure are you?

I'm sureI think soI'm guessing

It's £2 a mile, so 5 miles costs 5 × £2 = £10.

This is a common misconception

That treats the graph as if it starts at (0, 0). It doesn't: the fare is already £5 before the taxi moves. The £2 is added for each extra mile on top, so 5 miles costs £5 + 5 × £2 = £15.

5 miles costs £15, and the gradient means each extra mile adds £2 to the fare.

This matches the evidence

Exactly. Start at £5, add £2 for each of the 5 miles: £15. The gradient is the extra cost for each additional mile.

The gradient is the cost per mile overall, so the fare divided by the miles is always £2.

This is a common misconception

Test it: 5 miles costs £15, which is £3 per mile overall, not £2. Fare ÷ miles only equals the gradient when the line goes through the origin. Here the gradient is the extra £2 each additional mile adds.

Gradient = extra cost for each additional unit. Only a graph through the origin lets you find the total by multiplying.

Higher

Say it in context

What does the gradient mean?

The graph shows the total cost of a broadband contract, in pounds, against time in months. The gradient of the line is 20 for the first three months and 40 after that. Explain what each gradient tells you about the cost of the broadband. [3 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

First three months: the broadband costs £20 per month (a rate in pounds per month). (1 mark)
After three months: it costs £40 per month. (1 mark)
States that the monthly cost changes (goes up, or doubles) after three months. (1 mark)

Model answer

For the first three months the gradient is 20, so the broadband costs £20 per month. After that the gradient is 40, so it costs £40 per month. The monthly cost doubles once the first three months are over.

'The gradient is 20' earns nothing on its own. Turn it into the context's language: £20 per month.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A curve changes steepness at every point, so you estimate its gradient. And the space underneath has a meaning too.

What you need to know

A chord joins two points on a curve. Its gradient is the average rate of change between them.
The gradient at a single point is the gradient of the tangent there: a straight line that touches the curve at that point with the same gradient as the curve.
On a distance–time graph the gradient is the speed. On a speed–time graph it is the acceleration, and a negative gradient means deceleration.
The area between a speed–time graph and the time axis is the distance travelled. Estimate it for a curve by adding trapezia.
In a money context the gradient is a rate, such as pounds per month or the extra cost for each additional mile.

The big picture

A curved graph has a different gradient at every point, so you estimate it. A chord joining two points gives the average rate between them, and the closer the points, the better the estimate. For the rate at one point, draw the tangent there and find its gradient. Then say what the gradient means in context: speed on a distance–time graph, acceleration on a speed–time graph (negative means slowing down), and a rate such as pounds per month on a cost graph. The area under a speed–time graph is the distance travelled; for a curve, estimate it with trapezia, reading the scales rather than counting squares.

Key points

1The closer together the two points of a chord, the better its gradient estimates the gradient at a point.

2Where y gets smaller as x gets larger, the gradient is negative.

3A flat section means stopped on a distance–time graph, but steady speed on a speed–time graph.

4Read both axis scales: they are often different, so counting squares gives the wrong area.

5More, narrower trapezia give a better estimate of the area under a curve.

6If a cost graph doesn't pass through the origin, you can't find a total by multiplying the gradient by the number of units.

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Worked example

Problem

A tangent has been drawn to a curved distance–time graph at t = 4 s. It passes through the points (1, 0) and (7, 30), with time in seconds and distance in metres. Estimate the speed at t = 4 s.

⚠ Watch out

Calling a chord's gradient 'the speed at' a point. A chord gives the average speed over the gap between its two points; only the tangent gives the speed at one instant.

🧠

Memory hook

Chord for the journey, tangent for the moment. Two points far apart give the average; squeeze them together to catch the instant.

✓

Check yourself

A speed–time graph falls in a straight line from 12 m/s at t = 10 s to 4 m/s at t = 14 s. Find its gradient. What does it tell you about the motion?

Flashcards

(15)

What does the gradient of a chord on a curve give you?

The average rate of change between the chord's two points.

What is a tangent to a curve?

A straight line that touches the curve at one point and has the same gradient as the curve there.

How do you estimate the gradient of a curve at one particular point?

Draw the tangent at that point and work out the tangent's gradient.

How do you make a chord's gradient a better estimate of the gradient at a point?

Choose the two points closer together.

When is a gradient negative?

When y gets smaller as x gets larger: the line slopes down from left to right.

Distance–time graph: what does the gradient tell you?

The speed: how far the object goes in each unit of time.

Distance–time graph: what does a flat section mean?

No distance is being covered: the object isn't moving.

Speed–time graph: what does the gradient tell you, and what does a negative gradient mean?

The acceleration. A negative gradient means deceleration (slowing down).

Speed–time graph: what does a flat section mean?

The speed is steady, not zero.

What does the area between a speed–time graph and the time axis represent, and why?

The distance travelled, because speed × time = distance.

How do you estimate the area under a curved graph?

Split it into trapezia, work out each area and add them. More trapezia give a better estimate.

Area of a trapezium

½ × (sum of the parallel sides) × width.

Why read the scales instead of counting squares when finding an area?

The two axes often use different scales, so one square is not one unit of distance.

How should you explain a gradient in a real context?

As a rate in the context's own units, for example £20 per month.

A cost graph doesn't pass through the origin. What does its gradient mean?

The extra cost for each additional unit, not the cost per unit overall.

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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Squeeze a chord into a tangent

Estimate a distance from a curved speed–time graph

The £2-a-mile taxi

What does the gradient mean?

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