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Maths · Equations and graphs

Cubic equations

Draw and recognise cubic graphs, find their roots and turning points, and use a graph to estimate solutions of cubic equations.

  • Higher
  • 6 key terms
  • All boards
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Teacher resources

The teacher copies: slides with the questions built in, the answers, and anything else attached to this lesson for whoever is teaching it.

Student handouts

The same files the students see, to print or hand out.

Warm-up

Answer each one, then check.

  1. 1

    Work out \((-2)^3\).

    Show answerHide answer

    \(-8\)

  2. 2

    Factorise \(x^2 - 4\).

    Show answerHide answer

    \((x - 2)(x + 2)\)

  3. 3

    Work out \(2^3 - 4 \times 2\).

    Show answerHide answer

    \(0\)

  4. 4

    What is a root of a graph?

    Show answerHide answer

    Where \(y = 0\)

  5. 5

    What shape is \(y = x^2\)?

    Show answerHide answer

    A parabola

Learning Objectives

  1. 1Recognise the shape of a cubic graph.
  2. 2Complete a table and draw a cubic graph.
  3. 3Find roots by factorising or from the graph.
  4. 4Solve \(f(x) = k\) by drawing a horizontal line.

CUBIC GRAPH

A cubic graph has an \(x^3\) term and a smooth S-shaped curve. It can cross the x-axis up to three times.

With a positive \(x^3\) term the curve rises from bottom left to top right.

Table of Values

\(y = x^3 - 4x\).

  • \(y\)

    \(-3\): \(-15\). \(-2\): \(0\). \(-1\): \(3\). \(0\): \(0\) | \(-3\) | \(0\) | \(15\)

Roots of a Cubic

Find the roots of \(y = x^3 - 4x\).

Show the solutionHide the solution
  1. 1 Set \(y = 0\) \(x^3 - 4x = 0\)
  2. 2 Factorise \(x(x^2 - 4) = x(x - 2)(x + 2)\)
  3. 3 Solve \(x = 0,\ 2,\ -2\)

AnswerThe roots are \(x = -2\), \(0\) and \(2\).

Solving a Cubic from the Graph

Use the graph of \(y = x^3 - 4x\) to solve \(x^3 - 4x = 1\).

Show the solutionHide the solution
  1. 1 Draw The line \(y = 1\)
  2. 2 Read the three intersections \(x \approx -1.9\), \(-0.3\) and \(2.1\)
  3. 3 Check \(2.1^3 - 4 \times 2.1 = 0.861\), close to 1

Answer\(x \approx -1.9\), \(x \approx -0.3\) and \(x \approx 2.1\)

How Many Solutions?

How many solutions does \(x^3 - 4x = 5\) have?

Show the solutionHide the solution
  1. 1 Local maximum About 3.1, below 5
  2. 2 The line \(y = 5\) Meets the curve once, on the right

AnswerOne solution.

Common Mistakes

Draw carefully.

  • Sharp corners

    The curve is smooth: no straight segments or points.

  • Negative values

    Cube negatives carefully: \((-2)^3 = -8\).

  • Missing solutions

    A cubic can have up to three solutions.

  • Turning point values

    Read them from the graph or a fine table.

Cubic Table

Complete a table for \(y = x^3 - 3x\) for \(x = -2\) to \(2\). Sketch the graph. How many roots does it have?

1. Cube each x carefully.

2. Join with a smooth curve.

A good answer shows: \(y = -2, 2, 0, -2, 2\). The curve crosses the x-axis at \(x = -\sqrt{3}, 0, \sqrt{3}\): three roots.

Can I...?

  1. 1Recognise a cubic graph.
  2. 2Complete a table.
  3. 3Draw a smooth curve.
  4. 4Find roots.
  5. 5Identify turning points.
  6. 6Draw a line to solve an equation.
  7. 7Count the solutions.
  8. 8Avoid common mistakes.

Summary & Exam Focus

  • A cubic has an \(x^3\) term.
  • It can have up to three roots and two turning points.
  • Solve \(f(x) = k\) with the line \(y = k\).
  • Factorise to find exact roots.

Exam focus

Use the graph of \(y = x^3 - 4x\) to find estimates for the solutions of \(x^3 - 4x = 1\). (3 marks) (3 marks)

Draw the horizontal line \(y = 1\) and read all three intersections.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Cubic
An expression or equation with a highest power of \(x^3\).
Root
A solution of \(f(x) = 0\).
Local maximum
A turning point at the top of a hill.
Local minimum
A turning point at the bottom of a valley.
Intersection
Where two graphs meet.
Estimate
An approximate value from a graph.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Complete the table 2 marks Easier

Complete the table of values for \(y = x^3 - 4x\) for \(x = -3, -2, -1, 0, 1, 2, 3\).

Mark scheme — 2 marks available

  • At least 4 correct — M1
  • All correct — A1

Model answer

\(y = -15, 0, 3, 0, -3, 0, 15\)

2. Exam question Use the graph 3 marks Easier

The graph of \(y = x^3 - 4x\) is drawn on the grid. Use the graph to find estimates for the solutions of \(x^3 - 4x = 1\).

The graph of y equals x cubed minus 4x on a grid.

Mark scheme — 3 marks available

  • Draws \(y = 1\) — M1
  • Two correct estimates — A1
  • All three — A1

Model answer

Draw \(y = 1\). Solutions \(x \approx -1.9\), \(-0.3\) and \(2.1\). Accept \(-2.0\) to \(-1.8\), \(-0.4\) to \(-0.2\) and \(2.0\) to \(2.2\).

3. Exam question Solve 3 marks Easier

Factorise \(x^3 - 4x\) and hence solve \(x^3 - 4x = 0\).

Mark scheme — 3 marks available

  • \(x(x^2 - 4)\) — M1
  • \(x(x - 2)(x + 2)\) — A1
  • Three solutions — A1

Model answer

\(x(x - 2)(x + 2) = 0\), so \(x = 0\), \(x = 2\) or \(x = -2\).

4. Exam question Explain 2 marks Easier

Use the graph to explain why \(x^3 - 4x = 5\) has only one solution.

Mark scheme — 2 marks available

  • Line \(y = 5\) is above the maximum — M1
  • Only one intersection — C1

Model answer

The line \(y = 5\) is above the local maximum of the curve (about 3.1), so it meets the curve only once, on the right.

5. Exam question Work out 2 marks Easier

Work out the value of \(x^3 - 4x\) when \(x = 1.2\).

Mark scheme — 2 marks available

  • \(1.2^3 = 1.728\) — M1
  • \(-3.072\) — A1

Model answer

\(1.728 - 4.8 = -3.072\)

6. Exam question Describe 2 marks Easier

Describe the shape of the graph of \(y = x^3\) and write down its root.

Mark scheme — 2 marks available

  • S-shaped curve — B1
  • \(x = 0\) — B1

Model answer

A smooth S-shaped curve rising from bottom left to top right through the origin; the root is \(x = 0\).

7. Multiple choice 1 mark Easier

A cubic graph can cross the x-axis...

  1. A Only once
  2. B Exactly twice
  3. C Never
  4. D Up to three times Correct

Why: Up to three times.

8. Multiple choice 1 mark Core

\((-2)^3 =\)

  1. A \(-8\) Correct
  2. B \(8\)
  3. C \(-6\)
  4. D \(6\)

Why: \(-2 \times -2 \times -2 = -8\).

9. Multiple choice 1 mark Core

\(x^3 - 4x\) factorises to...

  1. A \(x(x - 4)\)
  2. B \((x - 2)^2\)
  3. C \(x(x - 2)(x + 2)\) Correct
  4. D \(x^2(x - 4)\)

Why: \(x(x - 2)(x + 2)\).

10. Multiple choice 1 mark Core

To solve \(x^3 - 4x = 1\) with the graph, draw...

  1. A \(x = 1\)
  2. B \(y = 1\) Correct
  3. C \(y = x\)
  4. D \(y = 0\)

Why: The line \(y = 1\).

11. Multiple choice 1 mark Core

Between roots of a cubic there is...

  1. A A turning point Correct
  2. B A gap
  3. C A straight line
  4. D Nothing

Why: A turning point.

12. Multiple choice 1 mark Stretch

The graph of \(y = x^3\) passes through...

  1. A \((1, 0)\)
  2. B \((0, 1)\)
  3. C \((0, 0)\) Correct
  4. D \((2, 6)\)

Why: The origin, and rises from bottom left to top right.