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Maths · More trigonometry

Transforming trigonometric graphs 2

Transform trigonometric graphs horizontally: translations \(y = f(x + a)\) and stretches \(y = f(ax)\), and find the period and equation from a graph.

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Teacher resources

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Student handouts

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Warm-up

Answer each one, then check.

  1. 1

    What is the period of \(y = \sin x\)?

    Show answerHide answer

    \(360^\circ\)

  2. 2

    Where is the first maximum of \(y = \sin x\)?

    Show answerHide answer

    \(x = 90^\circ\)

  3. 3

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

    Show answerHide answer

    Moves it up 2

  4. 4

    What is the period of \(y = \tan x\)?

    Show answerHide answer

    \(180^\circ\)

  5. 5

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

    Show answerHide answer

    Stretches it vertically by 3

Learning Objectives

  1. 1Describe and sketch \(y = \sin(x + a)\) and \(y = \cos(x - a)\).
  2. 2Describe and sketch \(y = \sin ax\) and \(y = \cos ax\).
  3. 3Find the period of \(y = \sin ax\).
  4. 4Solve equations such as \(\sin 2x = 0.5\).

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 \(\dfrac{1}{a}\).

Effects on the Graph

Read the change inside the brackets.

  • \(y = \sin(x - 30^\circ)\)

    Transformation: Translate right 30. Period: \(360^\circ\)

  • \(y = \sin(x + 60^\circ)\)

    Transformation: Translate left 60. Period: \(360^\circ\)

  • \(y = \sin 2x\)

    Transformation: Horizontal stretch, scale factor \(\tfrac{1}{2}\). Period: \(180^\circ\)

  • \(y = \cos 3x\)

    Transformation: Horizontal stretch, scale factor \(\tfrac{1}{3}\). Period: \(120^\circ\)

  • \(y = \sin\tfrac{x}{2}\)

    Transformation: Horizontal stretch, scale factor 2. Period: \(720^\circ\)

A Horizontal Translation

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

Show the solutionHide the solution
  1. 1 Plus inside the bracket Moves left by \(30^\circ\)
  2. 2 Maximum was at \(90^\circ\)
  3. 3 Now at \(90^\circ - 30^\circ = 60^\circ\)

AnswerA translation of \(30^\circ\) to the left; the first maximum is at \(x = 60^\circ\).

A Horizontal Stretch

State the period of \(y = \cos 3x\) and how many full waves there are between \(0^\circ\) and \(360^\circ\).

Show the solutionHide the solution
  1. 1 Period \(\dfrac{360^\circ}{3} = 120^\circ\)
  2. 2 Number of waves \(\dfrac{360}{120} = 3\)

AnswerThe period is \(120^\circ\), giving 3 full waves.

Solving sin 2x = 0.5

Solve \(\sin 2x = 0.5\) for \(0^\circ \le x \le 360^\circ\).

Show the solutionHide the solution
  1. 1 Range for \(2x\) \(0^\circ \le 2x \le 720^\circ\)
  2. 2 Solve for \(2x\) \(2x = 30^\circ,\ 150^\circ,\ 390^\circ,\ 510^\circ\)
  3. 3 Halve \(x = 15^\circ,\ 75^\circ,\ 195^\circ,\ 255^\circ\)

Answer\(x = 15^\circ,\ 75^\circ,\ 195^\circ,\ 255^\circ\)

Inside and Outside the Brackets

Outside the brackets: vertical

  • \(y = a\sin x\): 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 \(\tfrac{1}{a}\).
  • Does the opposite of what it looks like.

Period Detective

State the period of each graph. (a) \(y = \sin 4x\) (b) \(y = \cos\tfrac{x}{3}\) (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.

A good answer shows: (a) \(90^\circ\). (b) \(1080^\circ\). (c) \(90^\circ\) (tangent's normal period is \(180^\circ\), so divide by 2). First maximum of (a) at \(x = 22.5^\circ\).

Can I...?

  1. 1Describe y = sin(x + a).
  2. 2Describe y = sin(x - a).
  3. 3Describe y = sin ax.
  4. 4Find the period.
  5. 5Find a first maximum.
  6. 6Sketch a transformed graph.
  7. 7Solve sin 2x = k.
  8. 8Find a from a graph.

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 \(\tfrac{1}{a}\).
  • Period of \(\sin ax\) or \(\cos ax\): \(\dfrac{360^\circ}{a}\).
  • Solving \(\sin ax = k\): extend the range to \(360a\).

Exam focus

Solve \(\sin 2x = 0.5\) for \(0^\circ \le x \le 360^\circ\). (4 marks) (4 marks)

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

Key terms

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

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.

Questions and answers

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

1. Exam question Find 2 marks Easier

The graph shows \(y = \cos ax\) for \(0^\circ \le x \le 360^\circ\), where \(a\) is a positive whole number. Find the value of \(a\).

A cosine-shaped graph with three full waves between 0 and 360 degrees.

Mark scheme — 2 marks available

  • Three waves or period \(120^\circ\) — M1
  • 3 — A1

Model answer

\(a = 3\)

2. Exam question Describe 2 marks Easier

Describe the single transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = \sin(x - 40^\circ)\).

Mark scheme — 2 marks available

  • Translation — M1
  • \(40^\circ\) to the right — A1

Model answer

A translation of \(40^\circ\) to the right, that is by the vector \(\begin{pmatrix}40 \\ 0\end{pmatrix}\).

3. Exam question Write down 1 mark Easier

Write down the period of \(y = \sin 4x\).

Mark scheme — 1 mark available

  • 90 — B1

Model answer

\(90^\circ\)

4. Exam question Write down 2 marks Easier

Write down the coordinates of the first maximum point of \(y = \sin(x - 30^\circ)\) for \(x > 0\).

Mark scheme — 2 marks available

  • \(x = 120\) — B1
  • \(y = 1\) — B1

Model answer

\((120^\circ, 1)\)

5. Exam question Sketch 3 marks Easier

On a grid, sketch the graph of \(y = \sin 2x\) for \(0^\circ \le x \le 360^\circ\).

Mark scheme — 3 marks available

  • Two full waves — B1
  • Maximum 1 and minimum \(-1\) — B1
  • Correct roots or turning points — B1

Model answer

Two full sine waves: maxima of 1 at \(45^\circ\) and \(225^\circ\), minima of \(-1\) at \(135^\circ\) and \(315^\circ\), crossing the x-axis at \(0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ\).

6. Exam question Solve 4 marks Easier

Solve \(\sin 2x = 0.5\) for \(0^\circ \le x \le 360^\circ\).

Mark scheme — 4 marks available

  • \(2x = 30\) — M1
  • \(2x = 150\) — M1
  • Extends to \(390\) and \(510\) — M1
  • All four answers — A1

Model answer

\(x = 15^\circ,\ 75^\circ,\ 195^\circ,\ 255^\circ\)

7. Multiple choice 1 mark Easier

\(y = \sin(x + 20^\circ)\) is a translation of \(y = \sin x\)...

  1. A 20 left Correct
  2. B 20 right
  3. C 20 up
  4. D 20 down

Why: A plus inside the brackets moves the graph left.

8. Multiple choice 1 mark Core

The period of \(y = \cos 2x\) is...

  1. A \(360^\circ\)
  2. B \(90^\circ\)
  3. C \(180^\circ\) Correct
  4. D \(720^\circ\)

Why: \(360 \div 2 = 180\).

9. Multiple choice 1 mark Core

\(y = \sin\tfrac{x}{2}\) has period...

  1. A \(90^\circ\)
  2. B \(180^\circ\)
  3. C \(360^\circ\)
  4. D \(720^\circ\) Correct

Why: \(360 \div \tfrac{1}{2} = 720\).

10. Multiple choice 1 mark Core

How many full waves does \(y = \sin 3x\) have for \(0 \le x \le 360\)?

  1. A 2
  2. B 3 Correct
  3. C 6
  4. D 12

Why: The period is 120, so there are 3 full waves.

11. Multiple choice 1 mark Core

The first maximum of \(y = \sin(x - 50^\circ)\) is at...

  1. A \(40^\circ\)
  2. B \(90^\circ\)
  3. C \(140^\circ\) Correct
  4. D \(50^\circ\)

Why: \(90 + 50 = 140\).

12. Multiple choice 1 mark Stretch

To solve \(\sin 2x = 0.5\) for \(0 \le x \le 360\), the range for \(2x\) is...

  1. A 0 to 720 Correct
  2. B 0 to 360
  3. C 0 to 180
  4. D 0 to 90

Why: Double the range: 0 to 720.