Chapter 7 of 8

All 8 chapters
  1. 1Function notation, readunderstood
  2. 2Composite and inverse functions, readunderstood
  3. 3Cubic, reciprocal and circle graphs, readunderstood
  4. 4Exponential graphs, readunderstood
  5. 5Graphs of sin, cos and tan, readunderstood
  6. 6Moving graphs, readunderstood
  7. 7Gradients and areas under graphs, readunderstood
  8. 8Solving with graphs and trials, readunderstood

M4.10.7 Chapter 7

Gradients and areas under graphs

A cyclist speeds up all the time. How can I find her speed at one moment from the graph?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Estimate the gradient of a curve by drawing a tangent
  • Estimate the area under a curve using trapeziums
  • Explain what gradient and area mean on a speed time graph

A cyclist rides so that her distance after t seconds is d = t² metres. The graph of distance against time is a curve. It gets steeper, because she is speeding up.

For a straight line, the gradient is the rise divided by the run. On a distance time graph, the gradient is the speed: metres divided by seconds. A curve is steeper in some places than in others, so its gradient changes from point to point.

To find the gradient of a curve at a point, draw the tangent there. A tangent is a straight line that just touches the curve at that point, and runs the same way as the curve. The gradient of the curve at the point is the gradient of its tangent.

A graph with the x axis time (s) and the y axis distance (m). A curve rises from the origin and gets steeper. A straight line touches the curve at the point P at (3, 9). Two points on the straight line are named, A at (2, 3) and B at (4, 15). A dashed flat line runs from A to the right, and a dashed upright line runs from there up to B, to make a triangle.PAB123455101520250time (s)distance (m)
The tangent at P touches the curve at P only. The dashed triangle gives its rise and its run.

Worked example

Estimate the speed at 3 s, using the tangent through A (2, 3) and B (4, 15).

  1. Read two points on the tangent: A at (2, 3) and B at (4, 15).
  2. The rise is 15 − 3 = 12 m, and the run is 4 − 2 = 2 s.
  3. The gradient is 12 ÷ 2 = 6. So the speed at 3 s is about 6 m/s.

Answer: The speed is about 6 m/s.

On paper you draw the tangent with a ruler. Lay the ruler so that it touches the curve at your point. Turn it until the curve seems to leave the ruler at the same angle on both sides. Then draw the line, and pick two points far apart on it.

A chord is a straight line that joins two points of the curve. It is easy to draw, but it gives only an average gradient between its two ends. It is not the gradient at one point.

A graph with the x axis time (s) and the y axis distance (m). A curve rises from the origin and gets steeper. A solid straight line touches the curve at P at (3, 9). A dashed straight line, the chord, joins P to Q at (5, 25) and rises more steeply. A key under the graph names the solid line tangent at P and the dashed line chord PQ.PQ123455101520250time (s)distance (m)tangent at Pchord PQ
The chord PQ is steeper than the tangent at P. It gives the average speed from 3 s to 5 s, which is more than the speed at 3 s.

Now to areas. A tram goes at a steady 4 m/s for 5 s. The distance it travels is speed times time, which is 4 × 5 = 20 m. On a speed time graph this is the area of a rectangle.

See it three ways

With things

A tram goes at a steady 4 m/s for 5 s. How far does it go?

As a picture

A graph with the x axis time (s) and the y axis speed (m/s). A flat line at height 4 runs from time 0 to time 5. A dashed upright line at time 5 runs down from the end of the flat line to the x axis, closing a rectangle with the axes.1234561234560time (s)speed (m/s)

In words

Distance is speed times time. On a speed time graph that is the width times the height of the rectangle under the line, which is its area.

In symbols

distance = 4 × 5 = 20 m.

This works for any speed time graph, even a curve. The area under the graph, between the line and the time axis, is the distance travelled. For a curve we cut the area into thin strips, and treat the top of each strip as a straight line.

Each strip is then a trapezium, a shape with two parallel sides. Its area is the width times the average of the two parallel sides. If the sides are a and b and the width is w, the area is w × (a + b) ÷ 2. If one side is 0, the strip is a triangle, and the rule still works.

A graph with the x axis time (s) and the y axis speed (m/s). A curve shaped like a hill rises from the origin to a top near 9 at time 3, and falls back to the x axis at time 6. Seven dots mark the curve at each second. Upright lines at each second run from the x axis up to the curve, and dashed straight lines join the tops of neighbouring upright lines, making six strips.1234562468100time (s)speed (m/s)
The speed after each second. The six strips are 1 s wide. There are seven readings but only six strips.

Worked example

The speed of a runner is read each second. It is 0, 5, 8, 9, 8, 5 and 0 m/s. Estimate the distance, using trapeziums.

  1. Seven readings give six strips, each 1 s wide.
  2. The area of a strip is the width times the average of its two heights.
  3. The first strip: 1 × (0 + 5) ÷ 2 = 2.5.
  4. The others are 6.5, 8.5, 8.5, 6.5 and 2.5.
  5. Add the six areas: 2.5 + 6.5 + 8.5 + 8.5 + 6.5 + 2.5 = 35.

Answer: The distance is about 35 m.

This is an estimate. The curve here is a hill, and the straight tops of the strips lie just below it. So the estimate is a little too small. More, thinner strips give a better estimate.

A speed time graph has two meanings. Its gradient is the change of speed divided by the time. This is the acceleration, in metres per second per second, written m/s². The area under it is the distance travelled.

A graph with the x axis time (s) and the y axis speed (m/s). A straight line rises from the origin to the point A at (5, 10). A flat line runs from A to the point B at (15, 10). A straight line falls from B to the x axis at time 20. Dashed upright lines run down from A and from B to the x axis.AB5101520246810120time (s)speed (m/s)
A car speeds up for 5 s, goes at a steady speed, and then slows to a stop at 20 s.

Worked example

Use the graph of the car. Find its acceleration in the first 5 s, and the total distance it travels.

  1. The gradient of a speed time graph is the acceleration. In the first 5 s the speed rises from 0 to 10 m/s.
  2. So the acceleration is 10 ÷ 5 = 2 m/s².
  3. The area under the graph is a trapezium. Its parallel sides are 20 s and 10 s long, and their average is (20 + 10) ÷ 2 = 15.
  4. The height is 10 m/s, so the area is 15 × 10 = 150.
  5. So the car travels 150 m.

Answer: The acceleration is 2 m/s², and the distance is 150 m.

Try it

Squeeze the chord into a tangent

You need: a calculator, or paper and a pencil.

  1. On the curve d = t², take the point at t = 3, where d = 9.
  2. Join it to the points at t = 4, 3.5, 3.1 and 3.01. For each, work out the rise divided by the run.
  3. Look at the gradients. What number do they come close to?

What you will see: The gradients are 7, 6.5, 6.1 and 6.01. They get closer to 6, which is the gradient of the tangent at t = 3.

If you cannot try it, this is what happens.
A table with the heads to t (s) and gradient (m/s). Rows: 4 and 7; 3.5 and 6.5; 3.1 and 6.1; 3.01 and 6.01.to t (s)gradient (m/s)473.56.53.16.13.016.01

A wrong idea many people have

A chord gives the gradient of the curve at a point.

Why it feels right: A chord is a straight line that touches the curve at the point, and it is easy to draw.

The test that breaks it: For d = t², the chord from t = 3 to t = 5 has gradient (25 − 9) ÷ 2 = 8. But the tangent at t = 3 has gradient 6.

What is true: A chord joins two points and gives an average gradient between them. Only the tangent, which touches the curve at one point, gives the gradient at that point.

A wrong idea many people have

Seven readings make seven trapeziums.

Why it feels right: There are seven numbers in the list, so it seems that there are seven shapes.

The test that breaks it: The readings at 0, 1, 2, 3, 4, 5 and 6 s have only six gaps between them. The whole graph is 6 s wide. Seven strips would each be 6 ÷ 7 s wide, not 1 s.

What is true: Count the gaps, not the readings. With n readings there are n − 1 strips. The width of a strip is the gap between two readings.

Key words

tangent
a straight line that touches a curve at one point and runs the same way as the curve there.
chord
a straight line that joins two points of a curve.
trapezium
a shape with four sides, two of them parallel. Its area is the width times the average of the parallel sides.

Back to your guess

The answer is B: Draw a straight line that just touches the curve at that moment, and find its gradient.

On a distance time graph the gradient is the speed. A curve has a different gradient at every point. To find it at one moment, draw the tangent there. This is the straight line that just touches the curve. Then work out its gradient.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The straight line is the tangent to the curve d = t² at P, where t = 4. What is the speed at 4 s?

    A graph with the x axis time (s) and the y axis distance (m). A curve rises from the origin and gets steeper. A straight line touches the curve at the point P at (4, 16). Two points on the straight line are named, C at (3, 8) and D at (5, 24).PCD1234565101520250time (s)distance (m)
    Tap your answer.

    1. Not this one. You divided the distance 16 m by the time 4 s. That is the average speed since the start. The speed at 4 s is the gradient of the tangent. Go to M4.10.7. Try another.

    2. Yes, that is right. Use C at (3, 8) and D at (5, 24) on the tangent. The rise is 24 − 8 = 16 m and the run is 5 − 3 = 2 s. The gradient is 16 ÷ 2 = 8, so the speed is about 8 m/s.

    3. Not this one. The number 16 is the height of the curve at P, which is the distance in metres, not the speed. The speed is the gradient of the tangent. Go to M4.10.7. Try another.

    See the answer

    8 m/s

    Use C at (3, 8) and D at (5, 24) on the tangent. The rise is 24 − 8 = 16 m and the run is 5 − 3 = 2 s. The gradient is 16 ÷ 2 = 8, so the speed is about 8 m/s.

    • If you chose A: You divided the distance 16 m by the time 4 s. That is the average speed since the start. The speed at 4 s is the gradient of the tangent. Go to M4.10.7.
    • If you chose C: The number 16 is the height of the curve at P, which is the distance in metres, not the speed. The speed is the gradient of the tangent. Go to M4.10.7.

  2. The speed of a cyclist is 3, 7 and 9 m/s at 0 s, 2 s and 4 s. Use two trapeziums to estimate the distance.

    Tap your answer.

    1. Not this one. This is double the answer. Each trapezium is the width times half of the sum of its parallel sides, so remember to halve. Go to M4.10.7. Try another.

    2. Not this one. You added the three speeds. Each strip needs its own area: the width 2 s times the average of its two speeds. Go to M4.10.7. Try another.

    3. Not this one. You used a width of 1 s. The readings are 2 s apart, so each strip is 2 s wide. Go to M4.10.7. Try another.

    4. Yes, that is right. There are two strips, each 2 s wide. The first is 2 × (3 + 7) ÷ 2 = 10. The second is 2 × (7 + 9) ÷ 2 = 16. The total is 10 + 16 = 26.

    See the answer

    26 m

    There are two strips, each 2 s wide. The first is 2 × (3 + 7) ÷ 2 = 10. The second is 2 × (7 + 9) ÷ 2 = 16. The total is 10 + 16 = 26.

    • If you chose A: This is double the answer. Each trapezium is the width times half of the sum of its parallel sides, so remember to halve. Go to M4.10.7.
    • If you chose B: You added the three speeds. Each strip needs its own area: the width 2 s times the average of its two speeds. Go to M4.10.7.
    • If you chose C: You used a width of 1 s. The readings are 2 s apart, so each strip is 2 s wide. Go to M4.10.7.

  3. On a speed time graph, what does the area under the graph tell you?

    Tap your answer.

    1. Not this one. The acceleration is the gradient of a speed time graph, not its area. Go to M4.10.7. Try another.

    2. Not this one. The speed at the end is the height of the graph at that moment. The area is made of speed times time. Go to M4.10.7. Try another.

    3. Not this one. The time is read along the x axis. The area is made of speed times time, which is a distance. Go to M4.10.7. Try another.

    4. Yes, that is right. Speed times time is distance. The area under a speed time graph is made of speeds multiplied by times, so it is the distance travelled.

    See the answer

    The distance travelled

    Speed times time is distance. The area under a speed time graph is made of speeds multiplied by times, so it is the distance travelled.

    • If you chose A: The acceleration is the gradient of a speed time graph, not its area. Go to M4.10.7.
    • If you chose B: The speed at the end is the height of the graph at that moment. The area is made of speed times time. Go to M4.10.7.
    • If you chose C: The time is read along the x axis. The area is made of speed times time, which is a distance. Go to M4.10.7.

  4. A speed time graph is a straight line from (0, 6) to (4, 18), with speed in m/s and time in s. What is the acceleration?

    Tap your answer.

    1. Not this one. You used only the rise, 18 − 6. The acceleration is the gradient, so divide the rise by the run. Go to M4.10.7. Try another.

    2. Not this one. You divided the final speed 18 by the time 4. The line starts at 6, not 0, so the change of speed is 18 − 6. Go to M4.10.7. Try another.

    3. Yes, that is right. The acceleration is the gradient. The rise is 18 − 6 = 12 m/s, and the run is 4 − 0 = 4 s. So 12 ÷ 4 = 3 m/s².

    See the answer

    3 m/s²

    The acceleration is the gradient. The rise is 18 − 6 = 12 m/s, and the run is 4 − 0 = 4 s. So 12 ÷ 4 = 3 m/s².

    • If you chose A: You used only the rise, 18 − 6. The acceleration is the gradient, so divide the rise by the run. Go to M4.10.7.
    • If you chose B: You divided the final speed 18 by the time 4. The line starts at 6, not 0, so the change of speed is 18 − 6. Go to M4.10.7.

  5. Two straight lines pass through the point P of the curve d = t². Which line gives the gradient of the curve at P?

    A graph with the x axis time (s) and the y axis distance (m). A curve rises from the origin. Two straight lines pass through the point P at (2, 4). A key under the graph names the solid line line A and the dashed line line B. Line A joins P to the point (4, 16) on the curve. Line B touches the curve at P only.P123455101520250time (s)distance (m)line Aline B
    Tap your answer.

    1. Not this one. Line A joins P to another point of the curve, so it is a chord. It gives only an average gradient. Go to M4.10.7. Try another.

    2. Yes, that is right. Line B touches the curve at P only and runs the same way as the curve there. It is the tangent. Line A cuts the curve at a second point, so it is a chord.

    3. Not this one. Only one of them touches the curve at P alone. The other cuts the curve at a second point, so it shows an average gradient between two points. Go to M4.10.7. Try another.

    See the answer

    Line B

    Line B touches the curve at P only and runs the same way as the curve there. It is the tangent. Line A cuts the curve at a second point, so it is a chord.

    • If you chose A: Line A joins P to another point of the curve, so it is a chord. It gives only an average gradient. Go to M4.10.7.
    • If you chose C: Only one of them touches the curve at P alone. The other cuts the curve at a second point, so it shows an average gradient between two points. Go to M4.10.7.

  6. A curve is split into 5 strips of equal width, to use trapeziums. How many heights must you read from the graph?

    Tap your answer.

    1. Yes, that is right. Strips stand side by side, like the gaps in a fence: 5 gaps need 6 posts. The 5 strips have 6 edges, and a height is read at each edge. So there are 6 heights.

    2. Not this one. There are 5 strips, but each strip has two heights, one at each edge, and neighbouring strips share an edge. Five strips need 5 + 1 heights. Go to M4.10.7. Try another.

    3. Not this one. You counted two heights for every strip. But neighbouring strips share a height, so the edges are counted twice. Go to M4.10.7. Try another.

    See the answer

    6

    Strips stand side by side, like the gaps in a fence: 5 gaps need 6 posts. The 5 strips have 6 edges, and a height is read at each edge. So there are 6 heights.

    • If you chose B: There are 5 strips, but each strip has two heights, one at each edge, and neighbouring strips share an edge. Five strips need 5 + 1 heights. Go to M4.10.7.
    • If you chose C: You counted two heights for every strip. But neighbouring strips share a height, so the edges are counted twice. Go to M4.10.7.

  7. From an earlier chapter

    The curve y = f(x) has its lowest point at (1, 4). Where is the lowest point of y = f(x − 3)?

    Tap your answer.

    1. Not this one. You moved the point left. But a number taken away inside the brackets moves the graph to the right. Go to M4.10.6. Try another.

    2. Not this one. You added 3 to the y value. That is what f(x) + 3 does. With the 3 inside the brackets, the graph moves sideways. Go to M4.10.6. Try another.

    3. Yes, that is right. The lowest point is where the input of f is 1. We need x − 3 = 1, so x = 4. The height stays 4. The graph moves 3 to the right.

    See the answer

    (4, 4)

    The lowest point is where the input of f is 1. We need x − 3 = 1, so x = 4. The height stays 4. The graph moves 3 to the right.

    • If you chose A: You moved the point left. But a number taken away inside the brackets moves the graph to the right. Go to M4.10.6.
    • If you chose B: You added 3 to the y value. That is what f(x) + 3 does. With the 3 inside the brackets, the graph moves sideways. Go to M4.10.6.

  8. A tangent to a distance time curve passes through (2, 5) and (6, 25), with distance in m and time in s. What is the speed, in m/s?

    See the answer

    5

    The rise is 25 − 5 = 20 m and the run is 6 − 2 = 4 s. The gradient is 20 ÷ 4 = 5, so the speed is 5 m/s.

  9. A trapezium has parallel sides 5 m and 9 m long, and the distance between them is 4 m. Find its area, in m².

    See the answer

    28

    The area is the width times the average of the parallel sides. The average of 5 and 9 is 7, and 4 × 7 = 28.

Keep it

Copy this into your exercise book.

Gradient of a curve at a point: the gradient of the tangent. Area under a speed time graph: distance. Trapezium: width times the average of the two sides.

Draw: A curve with a tangent at one point, and a hill shaped curve cut into strips.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (956 KB). It opens on any phone.