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Maths · More trigonometry

Transforming trigonometric graphs 1

Transform trigonometric graphs vertically: stretches \(y = af(x)\), translations \(y = f(x) + a\) and reflections \(y = -f(x)\).

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  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What are the maximum and minimum of \(y = \sin x\)?

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    \(1\) and \(-1\)

  2. 2

    What is a reflection in the x-axis?

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    A flip over the x-axis; \(y\) changes sign

  3. 3

    What is a translation?

    Show answerHide answer

    A slide without turning

  4. 4

    What does the graph of \(y = \cos x\) start at?

    Show answerHide answer

    \(1\)

  5. 5

    What is the period of sine?

    Show answerHide answer

    \(360^\circ\)

Learning Objectives

  1. 1Describe and sketch \(y = a\sin x\) and \(y = a\cos x\).
  2. 2Describe and sketch \(y = \sin x + a\) and \(y = \cos x + a\).
  3. 3Describe and sketch \(y = -\sin x\) and \(y = -\cos x\).
  4. 4Find the equation of a transformed trigonometric graph from its maximum and minimum.

VERTICAL TRANSFORMATIONS

Changing the equation in a way that affects \(y\) moves or stretches the graph up and down.

\(y = a f(x)\) stretches by scale factor \(a\) in the y-direction. \(y = f(x) + a\) translates up by \(a\). \(y = -f(x)\) reflects in the x-axis.

Effects on the Graph

Read the equation to see the change.

  • \(y = \sin x\)

    Transformation: None. Maximum: \(1\). Minimum: \(-1\)

  • \(y = 3\sin x\)

    Transformation: Stretch, scale factor 3 in y. Maximum: \(3\). Minimum: \(-3\)

  • \(y = \cos x + 2\)

    Transformation: Translation up 2. Maximum: \(3\). Minimum: \(1\)

  • \(y = -\sin x\)

    Transformation: Reflection in the x-axis. Maximum: \(1\). Minimum: \(-1\)

  • \(y = 2\cos x - 1\)

    Transformation: Stretch by 2 then down 1. Maximum: \(1\). Minimum: \(-3\)

A Vertical Stretch

Sketch \(y = 3\sin x\) for \(0^\circ \le x \le 360^\circ\) and state its maximum and minimum.

Show the solutionHide the solution
  1. 1 Roots do not move \(x = 0^\circ,\ 180^\circ,\ 360^\circ\)
  2. 2 Multiply y-values by 3 Maximum \(3\) at \(90^\circ\)
  3. 3 Minimum \(-3\) at \(270^\circ\)

AnswerThe curve has the same shape as sine but is three times as tall: maximum 3, minimum \(-3\).

A Vertical Translation

State the maximum and minimum of \(y = \cos x + 2\) and its value when \(x = 0\).

Show the solutionHide the solution
  1. 1 Every point moves up 2 Maximum \(1 + 2 = 3\)
  2. 2 Minimum \(-1 + 2 = 1\)
  3. 3 At \(x = 0\) \(\cos 0 + 2 = 3\)

AnswerMaximum 3, minimum 1, and \(y = 3\) at \(x = 0\).

Finding the Equation

A graph has the shape of \(y = a\sin x + b\), with a maximum of 3 and a minimum of \(-1\). Find \(a\) and \(b\).

Show the solutionHide the solution
  1. 1 Middle line \(b = \dfrac{3 + (-1)}{2} = 1\)
  2. 2 Amplitude \(a = \dfrac{3 - (-1)}{2} = 2\)

Answer\(a = 2\) and \(b = 1\), so \(y = 2\sin x + 1\).

Stretch, Translation, Reflection

Changes the shape

  • \(y = a\sin x\): taller or shorter.
  • \(y = -\sin x\): upside down.
  • Roots stay where they are.

Moves the shape

  • \(y = \sin x + a\): up or down.
  • Roots move.
  • Maximum and minimum both move by \(a\).

Match the Graphs

Match each equation with its maximum and minimum. Equations: \(y = 2\sin x\), \(y = \sin x + 2\), \(y = -3\cos x\), \(y = \cos x - 1\). Options: (max 3, min 1), (max 2, min \(-2\)), (max 0, min \(-2\)), (max 3, min \(-3\)).

1. Work out the maximum first.

2. Then the minimum.

A good answer shows: \(y = 2\sin x\): (2, \(-2\)). \(y = \sin x + 2\): (3, 1). \(y = -3\cos x\): (3, \(-3\)). \(y = \cos x - 1\): (0, \(-2\)).

Can I...?

  1. 1Describe y = a sin x.
  2. 2Describe y = sin x + a.
  3. 3Describe y = -sin x.
  4. 4Find a maximum and a minimum.
  5. 5Sketch a transformed graph.
  6. 6Find the middle line.
  7. 7Find the amplitude.
  8. 8Write the equation from a graph.

Summary & Exam Focus

  • \(y = a f(x)\): vertical stretch, scale factor \(a\).
  • \(y = f(x) + a\): translation up by \(a\).
  • \(y = -f(x)\): reflection in the x-axis.
  • Roots stay put under a stretch or reflection.

Exam focus

The graph shows \(y = a\sin x + b\) for \(0^\circ \le x \le 360^\circ\). Find the values of \(a\) and \(b\). (2 marks) (2 marks)

\(b\) is the average of the maximum and minimum; \(a\) is half the difference.

Key terms

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

Amplitude
The height of the wave from the middle line to a maximum.
Stretch
A change that makes a graph taller or wider.
Translation
A slide without turning or flipping.
Reflection
A flip in a line such as the x-axis.
Middle line
The horizontal line halfway between maximum and minimum.
Scale factor
The number you multiply by.

Practice questions

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

  1. Question 1 Find 2 marks

    The graph shows \(y = a\sin x + b\) for \(0^\circ \le x \le 360^\circ\). Find the values of \(a\) and \(b\).

    A sine-shaped graph with maximum 3 at 90 degrees and minimum minus 1 at 270 degrees.
    Show answerHide answer

    Model answer

    \(a = 2\) and \(b = 1\)

    Mark scheme

    • \(a = 2\) — B1
    • \(b = 1\) — B1
  2. Question 2 Sketch 3 marks

    On a grid, sketch the graph of \(y = 3\sin x\) for \(0^\circ \le x \le 360^\circ\).

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    Model answer

    A sine curve through \((0, 0)\), \((180, 0)\) and \((360, 0)\) with a maximum of 3 at \(x = 90\) and a minimum of \(-3\) at \(x = 270\).

    Mark scheme

    • Correct shape — B1
    • Maximum 3 and minimum \(-3\) — B1
    • Roots correct — B1
  3. Question 3 Write down 2 marks

    Write down the maximum value and the minimum value of \(y = 2\cos x + 1\).

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    Model answer

    Maximum 3, minimum \(-1\).

    Mark scheme

    • 3 — B1
    • \(-1\) — B1
  4. Question 4 Describe 2 marks

    Describe the single transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = \sin x - 2\).

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    Model answer

    A translation of 2 units down, that is by the vector \(\begin{pmatrix}0 \\ -2\end{pmatrix}\).

    Mark scheme

    • Translation — M1
    • 2 down — A1
  5. Question 5 Describe 2 marks

    Describe the single transformation that maps the graph of \(y = \cos x\) onto the graph of \(y = -\cos x\).

    Show answerHide answer

    Model answer

    A reflection in the x-axis.

    Mark scheme

    • Reflection — M1
    • In the x-axis — A1
  6. Question 6 Write down 2 marks

    The graph of \(y = \cos x\) is transformed to the graph of \(y = 4\cos x\). Write down the coordinates of the point where the new graph crosses the y-axis, and its minimum value.

    Show answerHide answer

    Model answer

    \((0, 4)\); minimum \(-4\).

    Mark scheme

    • \((0, 4)\) — B1
    • \(-4\) — B1

Quick check

  1. The graph of \(y = 2\sin x\) has a maximum of...

    1. A1
    2. B2
    3. C3
    4. D4
    Show answerHide answer

    B: 2

    Multiply the maximum of sine by 2.

  2. \(y = \cos x + 3\) is a translation of \(y = \cos x\)...

    1. AUp 3
    2. BDown 3
    3. CLeft 3
    4. DRight 3
    Show answerHide answer

    A: Up 3

    Up by 3.

  3. \(y = -\sin x\) is a reflection of \(y = \sin x\) in the...

    1. Ay-axis
    2. BLine \(y = x\)
    3. Cx-axis
    4. DLine \(x = 90\)
    Show answerHide answer

    C: x-axis

    The x-axis: y-values change sign.

  4. The minimum of \(y = 3\cos x - 2\) is...

    1. A\(-3\)
    2. B\(-2\)
    3. C\(-1\)
    4. D\(-5\)
    Show answerHide answer

    D: \(-5\)

    \(3 \times (-1) - 2 = -5\).

  5. \(y = a\sin x + b\) has max 5 and min 1. Then \(b\) is...

    1. A2
    2. B3
    3. C4
    4. D5
    Show answerHide answer

    B: 3

    The middle line: \((5 + 1) \div 2 = 3\).

  6. Which transformation changes the roots of \(y = \sin x\)?

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

    C: \(y = \sin x + 1\)

    A translation up or down moves the graph off the x-axis, so the roots change.

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