Cubic equations.pptx

EDEXCEL GCSE MATHS · HIGHER

Cubic equations

Equations and graphs

Lesson 5 of 6

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EDEXCEL GCSE MATHS · HIGHER

Warm-up

Equations and graphs

Lesson 5 of 6

Answer each one, then check.

1

Work out (−2)³.

2

Factorise x² − 4.

3

Work out 2³ − 4 × 2.

4

What is a root of a graph?

5

What shape is y = x²?

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EDEXCEL GCSE MATHS · HIGHER

Warm-up - Answers

Equations and graphs

Lesson 5 of 6

1

Work out (−2)³.

−8

2

Factorise x² − 4.

(x − 2)(x + 2)

3

Work out 2³ − 4 × 2.

0

4

What is a root of a graph?

Where y = 0

5

What shape is y = x²?

A parabola

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EDEXCEL GCSE MATHS · HIGHER

Learning Objectives

Equations and graphs

Lesson 5 of 6

1

Recognise the shape of a cubic graph.

2

Complete a table and draw a cubic graph.

3

Find roots by factorising or from the graph.

4

Solve f(x) = k by drawing a horizontal line.

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EDEXCEL GCSE MATHS · HIGHER

Equations and graphs

Lesson 5 of 6

CUBIC GRAPH

A cubic graph has an x³ term and a smooth S-shaped curve. It can cross the x-axis up to three times.

With a positive x³ term the curve rises from bottom left to top right.

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EDEXCEL GCSE MATHS · HIGHER

A Cubic Graph

Equations and graphs

Lesson 5 of 6

Three roots, one maximum and one minimum.

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EDEXCEL GCSE MATHS · HIGHER

Table of Values

Equations and graphs

Lesson 5 of 6

y = x³ − 4x.

x

−3

−2

−1

1

y

−15

0

3

0 | −3 | 0 | 15

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EDEXCEL GCSE MATHS · HIGHER

Roots of a Cubic

Equations and graphs

Lesson 5 of 6

Find the roots of y = x³ − 4x.

1

Set y = 0

x³ − 4x = 0

2

Factorise

x(x² − 4) = x(x − 2)(x + 2)

3

Solve

x = 0, 2, −2

ANSWER

The roots are x = −2, 0 and 2.

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EDEXCEL GCSE MATHS · HIGHER

Solving a Cubic from the Graph

Equations and graphs

Lesson 5 of 6

Use the graph of y = x³ − 4x to solve x³ − 4x = 1.

1

Draw

The line y = 1

2

Read the three intersections

x ≈ −1.9, −0.3 and 2.1

3

Check

2.1³ − 4 × 2.1 = 0.861, close to 1

ANSWER

x ≈ −1.9, x ≈ −0.3 and x ≈ 2.1

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EDEXCEL GCSE MATHS · HIGHER

How Many Solutions?

Equations and graphs

Lesson 5 of 6

How many solutions does x³ − 4x = 5 have?

1

Local maximum

About 3.1, below 5

2

The line y = 5

Meets the curve once, on the right

ANSWER

One solution.

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EDEXCEL GCSE MATHS · HIGHER

Common Mistakes

Equations and graphs

Lesson 5 of 6

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.

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EDEXCEL GCSE MATHS · HIGHER

Key Terms

Equations and graphs

Lesson 5 of 6

Cubic

An expression or equation with a highest power of x³.

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.

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EDEXCEL GCSE MATHS · HIGHER

Equations and graphs

Lesson 5 of 6

YOUR TASK

Cubic Table

12 minutes

Complete a table for y = x³ − 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.

WHAT A GOOD ANSWER SHOWS

y = −2, 2, 0, −2, 2. The curve crosses the x-axis at x = −√3, 0, √3: three roots.

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EDEXCEL GCSE MATHS · HIGHER

Can I...?

Equations and graphs

Lesson 5 of 6

Recognise a cubic graph.

Complete a table.

Draw a smooth curve.

Find roots.

Identify turning points.

Draw a line to solve an equation.

Count the solutions.

Avoid common mistakes.

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EDEXCEL GCSE MATHS · HIGHER

Summary & Exam Focus

A cubic has an x³ 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 = x³ − 4x to find estimates for the solutions of x³ − 4x = 1. (3 marks)

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

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