Transforming trigonometric graphs 2.pptx

EDEXCEL GCSE MATHS · HIGHER

Transforming trigonometric graphs 2

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Lesson 9 of 9

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EDEXCEL GCSE MATHS · HIGHER

Warm-up

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Lesson 9 of 9

Answer each one, then check.

1

What is the period of y = sin x?

2

Where is the first maximum of y = sin x?

3

What does y = sin x + 2 do to the graph?

4

What is the period of y = tan x?

5

What does y = 3sin x do to the graph?

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Warm-up - Answers

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Lesson 9 of 9

1

What is the period of y = sin x?

360°

2

Where is the first maximum of y = sin x?

x = 90°

3

What does y = sin x + 2 do to the graph?

Moves it up 2

4

What is the period of y = tan x?

180°

5

What does y = 3sin x do to the graph?

Stretches it vertically by 3

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Learning Objectives

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Lesson 9 of 9

1

Describe and sketch y = sin (x + a) and y = cos (x − a).

2

Describe and sketch y = sin ax and y = cos ax.

3

Find the period of y = sin ax.

4

Solve equations such as sin 2x = 0.5.

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EDEXCEL GCSE MATHS · HIGHER

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Lesson 9 of 9

HORIZONTAL TRANSFORMATIONS

Changes inside the brackets move or stretch the graph left and right, and often do the opposite of what you expect.

y = f(x + a) translates left by a. y = f(x − a) translates right by a. y = f(ax) stretches horizontally with scale factor 1/a.

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Four Horizontal Transformations

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Lesson 9 of 9

The faint curve is y equals sine x.

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Effects on the Graph

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Lesson 9 of 9

Read the change inside the brackets.

Equation

Transformation

Period

y = sin (x − 30°)

Translate right 30

360°

y = sin (x + 60°)

Translate left 60

360°

y = sin 2x

Horizontal stretch, scale factor ½

180°

y = cos 3x

Horizontal stretch, scale factor ⅓

120°

y = sin x/2

Horizontal stretch, scale factor 2

720°

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A Horizontal Translation

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Lesson 9 of 9

The graph of y = sin x is translated to give y = sin (x + 30°). Describe the translation and find the first maximum.

1

Plus inside the bracket

Moves left by 30°

2

Maximum was at

90°

3

Now at

90°− 30°= 60°

ANSWER

A translation of 30° to the left; the first maximum is at x = 60°.

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A Horizontal Stretch

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Lesson 9 of 9

State the period of y = cos 3x and how many full waves there are between 0° and 360°.

1

Period

360°/3 = 120°

2

Number of waves

360/120 = 3

ANSWER

The period is 120°, giving 3 full waves.

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Solving sin 2x = 0.5

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Lesson 9 of 9

Solve sin 2x = 0.5 for 0° ≤ x ≤ 360°.

1

Range for 2x

0° ≤ 2x ≤ 720°

2

Solve for 2x

2x = 30°, 150°, 390°, 510°

3

Halve

x = 15°, 75°, 195°, 255°

ANSWER

x = 15°, 75°, 195°, 255°

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EDEXCEL GCSE MATHS · HIGHER

Inside and Outside the Brackets

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Lesson 9 of 9

OUTSIDE THE BRACKETS: VERTICAL

• y = asin x: stretch in y.

• y = sin x + a: translate up.

• Does what it looks like.

INSIDE THE BRACKETS: HORIZONTAL

• y = sin (x + a): translate left.

• y = sin ax: stretch in x by 1/a.

• Does the opposite of what it looks like.

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Key Terms

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Lesson 9 of 9

Period

The length after which the graph repeats.

Horizontal stretch

A stretch that changes the x-values.

Translation

A slide left, right, up or down.

Scale factor

The number the x-values are multiplied by.

Full wave

One complete cycle of the graph.

Phase shift

A horizontal translation of a trig graph.

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Lesson 9 of 9

YOUR TASK

Period Detective

12 minutes

State the period of each graph. (a) y = sin 4x (b) y = cos x/3 (c) y = tan 2x. Then say where the first maximum of (a) is.

1

Divide the normal period by the number.

2

Divide the first maximum position in the same way.

WHAT A GOOD ANSWER SHOWS

(a) 90°. (b) 1080°. (c) 90° (tangent's normal period is 180°, so divide by 2). First maximum of (a) at x = 22.5°.

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Can I...?

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Lesson 9 of 9

Describe y = sin(x + a).

Describe y = sin(x - a).

Describe y = sin ax.

Find the period.

Find a first maximum.

Sketch a transformed graph.

Solve sin 2x = k.

Find a from a graph.

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Summary & Exam Focus

y = f(x + a): translate left by a. y = f(x − a): translate right.

y = f(ax): stretch in x by scale factor 1/a.

Period of sin ax or cos ax: 360°/a.

Solving sin ax = k: extend the range to 360a.

EXAM FOCUS

Solve sin 2x = 0.5 for 0° ≤ x ≤ 360°. (4 marks)

Find the range for the whole bracket first. Then solve, then divide.

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