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Graphical Solutions

Graphical Solutions

Gradients of curves: tangents

Table, Graph And Tangent Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. The gradient of a curve
  4. Ways they test it
  5. Mistakes to avoid
  6. Summary
  7. Practice
  8. Can I do this?
  9. Quiz

In this note

Syllabus 2.9/2.10: gradients of curves
  • 2.10Draw a tangent at a point and use it to estimate the gradient of a curve.
  • 2.9Interpret a gradient as a rate of change (speed from a distance-time graph).
  • 2.10Know the gradient is zero at a turning point.

Before you start: gradient of a line and Solving equations with graphs. Exact gradients (0580 Extended): Differentiation.

Formula bank

GradientLearn it\(\dfrac{\text{change in }y}{\text{change in }x}=\dfrac{y_2-y_1}{x_2-x_1}\)use two points ON THE TANGENT, far apart.Use it when estimating a curve's gradient.
Rate of changeLearn itgradient = y-units per x-unitdistance-time: speed; speed-time: acceleration.Use it when a graph in context.

The gradient of a curve

A straight line has the same gradient everywhere; a curve's gradient keeps changing. The gradient AT a point is the gradient of the tangent there - the straight line that just touches the curve at that point, going in the same direction.

tangent to y = x² at x = 224-4-224681012xy(2, 4)(1, 0)(3, 8)
Tangent to y = x² at (2, 4).

Two points on the tangent: (1, 0) and (3, 8). Gradient \(=\dfrac{8-0}{3-1}=4\).

General methodHow to estimate the gradient of a curve at a point

  1. Place a ruler so it touches the curve at the point, with the curve on ONE side of the ruler near the point.
  2. Draw a long straight line (the tangent).
  3. Choose two points ON THE TANGENT, far apart, where it crosses grid lines.
  4. Gradient = (change in y) ÷ (change in x). Check the sign: uphill +, downhill −.
  5. In context, add units: y-units per x-unit.
Use points on your tangent, not on the curve. A big triangle gives a more accurate answer.
The curve is …Gradient
going up (left to right)positive
going downnegative
at a turning point0 (horizontal tangent)

Try it yourself

1
A tangent passes through (0, 1) and (4, 9). What is its gradient?
Show answer

Answer 2

2
What is the gradient of a curve at a minimum point?
Show answer

Answer 0

Ways they test it

Type 1 Estimate a gradient

"By drawing a suitable tangent, estimate the gradient of the curve where x = 3." (3 marks)

Spot it 'tangent' and 'estimate'.

  1. Draw the tangent.
  2. Big triangle on it.
  3. Rise ÷ run (accepted within a range).

For example Tangent to y = x² − 2x − 3 at (3, 0) through (5, 8)

  1. \(\dfrac{8-0}{5-3}\)

Answer ≈ 4

Type 2 Rate in context

"Estimate the speed of the car at t = 3 s." (2 marks)

Spot it a distance-time CURVE.

  1. Tangent at t = 3.
  2. Gradient = speed, with units.

For example Tangent through (0, 0) and (6, 24)

  1. \(\dfrac{24}{6}\)

Answer 4 m/s

Type 3 Zero gradient

"Write down the coordinates of the point where the gradient is 0."

Spot it 'gradient is zero'.

  1. It is the turning point: read it off the graph.

Mistakes to avoid

  • ✗ A line that cuts through the curve at the point✓ a tangent that just touches itIt must follow the curve's direction.
  • ✗ A tiny triangle✓ a big triangleMore accurate.
  • ✗ Dividing the point's y by its x✓ rise ÷ run on the tangentGradient is steepness, not position.

Summary

  • Gradient of a curve at a point = gradient of the tangent there.
  • Use a big triangle on the tangent: rise ÷ run.
  • Zero at a turning point.
  • In context the gradient is a rate (e.g. speed), with units.

If you see …

  • "by drawing a suitable tangent"touch the curve at the point, then rise ÷ run.
  • a distance-time curvethe gradient is the speed.

Practice

Practice: easy, medium, hard

Easy

1
A tangent passes through (1, 0) and (3, 8). Gradient?
Show answer

Answer 4

2
A tangent passes through (0, 6) and (3, 0). Gradient?
Hint
Downhill: negative.
Show answer

Answer −2

3
The gradient of a distance-time graph is the … ? Type a word.
Show answer

Answer speed

Medium

1
A tangent to \(y=x^{2}-2x-3\) at (3, 0) also passes through (5, 8). Estimate the gradient at x = 3.
Guide me

What is it asking? a gradient

What do we know? two points on the tangent

Plan \(\tfrac{8-0}{5-3}\)

Show working
  1. \(\dfrac{8}{2}=4\)

Answer 4

2
A distance-time curve (metres, seconds) has a tangent at t = 3 through (0, 0) and (6, 24). Estimate the speed at t = 3, in m/s.
Guide me

What is it asking? a speed

What do we know? gradient of the tangent

Plan 24 ÷ 6

Show working
  1. \(\dfrac{24-0}{6-0}=4\) m/s

Answer 4

3
\(y=x^{2}-2x-3\). At which x is the gradient zero?
Guide me

What is it asking? an x-value

What do we know? the turning point

Plan halfway between the roots −1 and 3

Show working
  1. x = 1

Answer 1

Hard

1
Estimate the gradient of \(y=2^{x}\) at \(x=2\) (the tangent at (2, 4) passes close to (1, 1.2) and (3, 6.8)).
Guide me

What is it asking? a gradient

What do we know? two points on the tangent

Plan \(\tfrac{6.8-1.2}{3-1}\)

Show working
  1. \(\dfrac{5.6}{2}=2.8\)

Answer 2.8

PartAnswer / workingMarksGuidance
tangent at (2, 4)B1
rise ÷ runM1
2.5 to 3.1A1
Total: 3 marks
2
For \(y=x^{2}\), the tangent from P(0, −4) touches the curve at (2, 4). Find the gradient of this tangent.
Guide me

What is it asking? a gradient

What do we know? it passes through (0, −4) and (2, 4)

Plan \(\tfrac{4-(-4)}{2-0}\)

Show working
  1. \(\dfrac{8}{2}=4\)

Answer 4

PartAnswer / workingMarksGuidance
\(\dfrac{8}{2}\)M1
4A1
Total: 2 marks
3
A curve rises, reaches a maximum at (3, 7) and falls. What is the gradient at x = 3?
Guide me

What is it asking? a gradient

What do we know? a turning point

Plan horizontal tangent

Show working
  1. 0

Answer 0

PartAnswer / workingMarksGuidance
0B1
Total: 1 mark

Can I do this?

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Quiz

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