Cubic and reciprocal graphs.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Cubic and reciprocal graphs

Graphs

Lesson 7 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Graphs

Lesson 7 of 8

Answer each one, then check.

1

Work out (−2)³.

2

Last lesson: what shape is the graph of y = x²?

3

Work out 12 ÷ 0.5.

4

What is the reciprocal of 3?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Graphs

Lesson 7 of 8

1

Work out (−2)³.

−8

2

Last lesson: what shape is the graph of y = x²?

A U-shaped parabola

3

Work out 12 ÷ 0.5.

24

4

What is the reciprocal of 3?

⅓

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Graphs

Lesson 7 of 8

1

Complete a table of values for a cubic and plot its graph.

2

Complete a table of values for a reciprocal and plot its graph.

3

Recognise and sketch linear, quadratic, cubic and reciprocal graphs.

4

Use a graph to solve equations.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Cubic Graphs

Graphs

Lesson 7 of 8

A cubic has an x³ term and no higher power.

y = x³

Passes through the origin: (−2, −8), (−1, −1), (0, 0), (1, 1), (2, 8).

S-shape

Positive x³: rises from bottom left to top right. Negative x³: falls from top left to bottom right.

Up to three roots

y = x³ − 4x = x(x − 2)(x + 2) crosses the x-axis at −2, 0 and 2.

Cube negatives carefully

(−3)³ = −27, so x³ − 4x at x = −3 is −27 + 12 = −15.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Table of Values for y = x³ − 4x

Graphs

Lesson 7 of 8

x

x³

−4x

y

−3

−27

12

−15

−2

−8

8

0

−1

−1

4

3

0

0

0

0

1

1

−4

−3

2

8

−8

0

3

27

−12

15

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reciprocal Graphs

Graphs

Lesson 7 of 8

A reciprocal graph has the form y = k/x.

No value at x = 0

You cannot divide by 0, so there is no point on the y-axis.

Two separate branches

For positive k, one branch is in the top right and one in the bottom left.

Asymptotes

The curve gets closer and closer to both axes but never touches them.

Negative k

The branches are in the top left and bottom right instead.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Plotting y = 12/x

Graphs

Lesson 7 of 8

Complete a table of values for y = 12/x for x = 1, 2, 3, 4, 6, 12, and describe the graph.

1

Divide 12 by each value

12, 6, 4, 3, 2, 1

2

Plot and join with a smooth curve

It falls steeply, then levels off

3

Negative values

x = −1 gives −12, x = −2 gives −6: the same shape, upside down, in the bottom left

ANSWER

y = 12, 6, 4, 3, 2, 1. The curve never touches either axis.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Four Graph Shapes

Graphs

Lesson 7 of 8

Linear, quadratic, cubic and reciprocal.

Know these four shapes by sight. The highest power of x tells you which: power 1 is a straight line, power 2 a parabola, power 3 an S-shaped cubic, and x on the bottom of a fraction a reciprocal with two branches.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Match the Equation to the Graph

Graphs

Lesson 7 of 8

Equation

Shape of graph

y = 3 − 2x

A

Two branches, top right and bottom left

1

y = x² − 1

B

Two branches, top left and bottom right

2

y = −x² + 4

C

Straight line sloping down

3

y = x³ + 1

D

U-shaped parabola

4

y = 5/x

E

∩-shaped parabola

5

y = −5/x

F

S-shaped curve rising to the right

6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Graphs

Lesson 7 of 8

Cubic

An expression whose highest power of x is x³.

Reciprocal graph

A graph of the form y = k/x.

Asymptote

A line that a curve gets closer and closer to but never touches.

Root

An x-value where a graph crosses the x-axis.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Graphs

Lesson 7 of 8

YOUR TASK

Graph Sketch Race

10 minutes

Your teacher calls out an equation. Sketch its graph on a mini-whiteboard in 20 seconds: y = 4, y = x³, y = −x², y = 2/x, y = 2x − 3, y = −x³, x = −1, y = x² + 2, y = −3/x.

1

Decide the type from the highest power.

2

Decide the direction from the sign.

3

Mark any intercepts.

WHAT A GOOD ANSWER SHOWS

Horizontal line; S-shape rising; ∩-shape through the origin; branches top-right and bottom-left; straight line through (0, −3), gradient 2; S-shape falling; vertical line; U-shape with minimum (0, 2); branches top-left and bottom-right.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Graphs

Lesson 7 of 8

Complete a table of values for a cubic.

Plot a cubic graph.

Complete a table of values for a reciprocal graph.

Plot a reciprocal graph.

Recognise linear, quadratic, cubic and reciprocal graphs.

Explain why y = k/x has no point at x = 0.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Cubic: S-shaped, up to three roots.

Reciprocal y = k/x: two branches, never touching the axes.

The highest power of x tells you the type of graph.

EXAM FOCUS

Match each equation to one of the graphs A, B, C and D: y = 3 − 2x, y = x² − 1, y = x³, y = 5/x. (2 marks)

In a matching question, start with the equations you are surest of - the straight line and the reciprocal are usually easiest - and eliminate.

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