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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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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.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Complete the table 2 marks

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

    Show answerHide answer

    Model answer

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

    Mark scheme

    • At least 4 correct — M1
    • All correct — A1
  2. Question 2 Use the graph 3 marks

    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.
    Show answerHide answer

    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 \(2.2\).

    Mark scheme

    • Draws \(y = 1\) — M1
    • Two correct estimates — A1
    • All three — A1
  3. Question 3 Solve 3 marks

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

    Show answerHide answer

    Model answer

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

    Mark scheme

    • \(x(x^2 - 4)\) — M1
    • \(x(x - 2)(x + 2)\) — A1
    • Three solutions — A1
  4. Question 4 Explain 2 marks

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

    Show answerHide answer

    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.

    Mark scheme

    • Line \(y = 5\) is above the maximum — M1
    • Only one intersection — C1
  5. Question 5 Work out 2 marks

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

    Show answerHide answer

    Model answer

    \(1.728 - 4.8 = -3.072\)

    Mark scheme

    • \(1.2^3 = 1.728\) — M1
    • \(-3.072\) — A1
  6. Question 6 Describe 2 marks

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

    Show answerHide answer

    Model answer

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

    Mark scheme

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

Quick check

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

    1. AOnly once
    2. BExactly twice
    3. CNever
    4. DUp to three times
    Show answerHide answer

    D: Up to three times

    Up to three times.

  2. \((-2)^3 =\)

    1. A\(-8\)
    2. B\(8\)
    3. C\(-6\)
    4. D\(6\)
    Show answerHide answer

    A: \(-8\)

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

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

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

    C: \(x(x - 2)(x + 2)\)

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

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

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

    B: \(y = 1\)

    The line \(y = 1\).

  5. Between roots of a cubic there is...

    1. AA turning point
    2. BA gap
    3. CA straight line
    4. DNothing
    Show answerHide answer

    A: A turning point

    A turning point.

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

    1. A\((1, 0)\)
    2. B\((0, 1)\)
    3. C\((0, 0)\)
    4. D\((2, 6)\)
    Show answerHide answer

    C: \((0, 0)\)

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

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