National 5 Maths
Trigonometric Graphs

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Course content

• Working with the graphs of trigonometric functions:
  • basic graphs
  • amplitude
  • vertical translation
  • multiple angle
  • phase angle

Textbook page references

• Zeta National 5+ Maths pp.208-214
• TeeJay Maths Book N5 pp.156-169
• Leckie National 5 Maths pp.249-271

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Basic trig graphs

• Period = 360° (width of one wave)
• Amplitude = 1 (half the height)
• Roots: 0°, 180°, 360°
• Maximum turning point (90,1)
• Minimum turning point (270,–1)

 

• Period = 360°
• Amplitude = 1
• Roots: 90°, 270°
• Maximum t.p. (0,1) and (360,1)
• Minimum turning point (180,–1)
• It's the sine graph moved left 90°

 

• Period = 180°
• Amplitude is undefined
• Roots: 0°, 180°, 360°
• No maximums or minimums
• "Asymptotes" at \(x\) = 90° and 270°

Change of amplitude

Now we will use the sine function to look at change of amplitude and the other three transformation types that you need to understand for Nat 5 Maths.

Example:  \(y=5\small \ \normalsize sin\ x\)

• Amplitude 5 (vertical stretch: \(\times\)5)
• Maximum turning point (90,5)
• Minimum turning point (270,–5)

Example:  \(y=\small \frac{1}{2}\tiny \ \normalsize\normalsize sin\ x \)

• Amplitude \( \small \frac{1}{2}\) (vertical shrink: ÷2)
• Maximum turning point (90,\( \small \frac{1}{2}) \)
• Minimum turning point (270,\( –\small \frac{1}{2} \))

Example:  \(y=-sin\ x\)

• Amplitude remains 1
• Reflects in the \(x\)-axis (upside-down)
• Maximum turning point (270,1)
• Minimum turning point (90,–1)

Example:  \(y=-3\small \ \normalsize sin\ x \)

• Amplitude 3 (vertical stretch: \(\times\)3)
• The graph also reflects in the \(x\)-axis
• Maximum turning point (270,3)
• Minimum turning point (90,–3)

Vertical translation

Example:  \( y=sin\ x+2 \)

• The basic graph moves up 2
• Maximum turning point (90,3)
• Minimum turning point (270,1)

Example:  \( y=sin\ x-1 \)

• The basic graph moves down 1
• Maximum turning point (90,0)
• Minimum turning point (270,–2)

Vertical transformations

Now let's combine a change of amplitude and a vertical translation.

Example:  \(y=2\small \ \normalsize sin\ x+3\)

• First, the amplitude doubles
• Then that graph moves up 3
• Maximum turning point (90,5)
• Minimum turning point (270,1)

Example:  \(y=\small \frac{1}{2}\tiny \ \normalsize \normalsize sin\ x-1\)

• First, the amplitude halves
• Then that graph moves down 1
• Maximum turning point (90,\( \small -\frac{1}{2} \normalsize \))
• Minimum turning point (270,\( \small -\frac{3}{2} \normalsize \))

Example:  \(y=-2\small \ \normalsize sin\ x+5\)

• Amplitude \( \times \)2 (double-height)
• Reflect in the \(x\)-axis (upside-down)
• Then move the graph up 5
• Maximum turning point (270,7)
• Minimum turning point (90,3)

Multiple angle

Example:  \(y=sin\ 2x\)

• Each \(x\)-value divides by 2
• Horizontal shrink: half-width
• The period is 360 ÷ 2 = 180°
• There are 2 cycles from 0° to 360°
• Roots 0°, 90°, 180°, ...

Example:  \(y=sin\ \small \frac{1}{3}\normalsize x\)

• Each \(x\)-value divides by \(\small \frac{1}{3}\), i.e. \(\times\)3
• Horizontal stretch: triple-width
• The period is 360 ÷ \(\small \frac{1}{3}\) = 1080°
• There is \(\small \frac{1}{3}\) of a cycle from 0° to 360°
• Roots 0°, 540°, 1080°, ...

Example:  \(y=sin\ (-4x)\)

• Each \(x\)-value divides by –4
• Horizontal shrink: quarter-width
• Also a reflection in the y-axis
• The period is 360 ÷ 4 = 90°
• There are 4 cycles from 0° to 360°
• Roots 0°, 45°, 90°, 135°, ...
• The SQA has never set a question with a negative multiple angle.

Phase angle

Example:  \(y=sin\ (x-60)°\)

• Horizontal translation: right 60°
• Roots 60°, 240°, 420°, ...

Example:  \(y=sin\ (x+45)°\)

• Horizontal translation: left 45°
• Roots –45°, 135°, 315°, ...

A useful way to remember:

• MINUS and RIGHT: 5 letters each
• PLUS and LEFT: 4 letters each

Horizontal transformations

Now let's combine a multiple angle and a phase angle. An unexpected thing happens: the phase angle always acts before the multiple angle. PM, not MP!

Example:  \(y=sin\ (2x+60)° \)

• First, we translate left 60°
• Then each \(x\)-value divides by 2
• The period is 360 ÷ 2 = 180°
• Roots: 60°, 150°, 240°, 330°, ...

Example:  \(y=sin\ (\small \frac{1}{2}\normalsize x-30)° \)

• First, we translate right 30°
• Then each \(x\)-value divides by \( \frac{1}{2} \)
• The period is 360 ÷ \( \frac{1}{2} \) = 720°
• Roots: 60°, 420°, 780°, ...

2D transformations

Finally, we need to be able to combine horizontal and vertical transformations.

Example:  \(y=3\small \ \normalsize sin\ (x+25)° \)

• Horizontal:
  • Translate left 25
  • No multiple angle here
  • The period remains 360°
• Vertical:
  • Stretch: amplitude becomes 3
  • No reflection or translation
  • Max. turning point (65,3)
  • Min. turning point (245,–3)

Example:  \(y=-2\small \ \normalsize sin\ 3x - 4 \)

• Horizontal:
  • No phase angle here
  • All \(x\)-values divide by 3
  • The period is 120°
• Vertical:
  • Stretch: amplitude becomes 2
  • Reflect in the \(x\)-axis
  • Then translate down 4
  • Max. value: –2\(\times\)–1 – 4 = –2
  • Max. turning point (90,–2)
  • Min. value: –2\(\times\)1 – 4 = –6
  • Min. turning point (30,–6)

Example:  \(y=5\small \ \normalsize sin\ (2x-30)°+1 \)

• Horizontal:
  • Translate right 30°
  • Then \(x\)-values divide by 2
  • The period is 180°
• Vertical:
  • Stretch: amplitude becomes 5
  • Then translate up 1
  • Max. value: 5\(\times\)1 + 1 = 6
  • Max. turning point (60,6)
  • Min. value: 5\(\times\)–1 + 1 = –4
  • Min. turning point (150,–4)

So far, the SQA have only included two or three of the four transformation types in their questions, so that last example should have over-prepared you!

All past paper questions so far have involved a given graph, from which you need to identify either the equation or the coordinates of a turning point. Some past paper questions are explained below.

Example 1 (non-calculator)

SQA National 5 Maths 2015 P1 Q6

Part of the graph of \(y=a\,sin\,bx^\circ\) is shown in the diagram.

State the values of \(a\) and \(b\small.\)

Show answer

Example 2 (non-calculator)

SQA National 5 Maths 2018 P1 Q6

Part of the graph of \(y=a\,cos\,bx^\circ\) is shown in the diagram.

State the values of \(a\) and \(b\small.\)

Show answer

Recommended student book

Zeta Maths: National 5 Maths Textbook 
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Example 3 (non-calculator)

SQA National 5 Maths 2019 P1 Q13

Part of the graph of \(y=3\,cos(x+45)^\circ\) is shown in the diagram.

The graph has a minimum turning point at A. State the coordinates of A.

Show answer

Example 4 (non-calculator)

SQA National 5 Maths 2021 P1 Q13

The graph of \(y=a\,cos\,x^\circ +b\small,\) \(0\leq x\leq 360\small,\) is given.

State the values of \(a\) and \(b\small.\)

Show answer

N5 Maths practice papers

Non-calculator papers and solutions 
Calculator papers and solutions 

Example 5 (non-calculator)

SQA National 5 Maths 2022 P1 Q8

Part of the graph of \(y=a\,sin\,bx^\circ\) is shown in the diagram.

(a)  State the value of \(a\small.\)
(b)  State the value of \(b\small.\)

Show answer

Example 6 (non-calculator)

SQA National 5 Maths 2023 P1 Q13

Part of the graph of \(y=cos(x+a)^\circ + b\) is shown.

(a)  State the value of \(a\small.\)
(b)  State the value of \(b\small.\)

Show answer

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Past paper questions

• All past paper questions by topic

• 2013 Specimen Paper 2 Q10(a) • 2014 Paper 1 Q10 • 2015 Paper 1 Q6 • 2015 Paper 1 Q9 (ordering) • 2018 Paper 1 Q6 • 2019 Paper 1 Q13 • 2021 Paper 1 Q13 • 2021 Paper 1 Q16 (with functions) • 2022 Paper 1 Q8 • 2023 Paper 1 Q13 • 2024 Paper 1 Q8

Other great resources

Videos - Mr Graham Maths 1. Amplitude 2. Period 3. Vertical translation 4. Phase angle

Videos - Mr Hamilton Online 1. Amplitude and period 2. Vertical shift 3. Horizontal shift 4. Summary: sin and cos 5. Trig graph exam questions

Video - YouKenMaths

Video - Tutorlene

Notes and videos - Mistercorzi

Notes and examples - Mearns Maths

PowerPoint - MathsRevision.com

Interactive graphing tool - NCTM

• Detailed notes - BBC Bitesize • Test yourself - BBC Bitesize

Notes - Maths4Scotland

Notes - National5.com

Notes - D R Turnbull

Lesson notes - Maths 777 1. Basic, amplitude and period 2. Translations in either direction

Examples - Maths Mutt 1. Sketching trig graphs 2. Amplitude and period

Practice questions - Maths Hunter

Worksheets - Starting Points Maths 1. Maximum and minimum values 2. Horizontal transformations 3. Complete workbook

Exercises - Larkhall Academy Pages 22-29 Ex 1-3

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