National 5 Maths
Trigonometric Graphs

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

National 5 Maths - trig graphs - basic graph for sine function, y = sin x

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)

National 5 Maths - trig graphs - basic graph for cosine function, y = cos x

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°

National 5 Maths - trig graphs - basic graph for tangent function, y = tan x

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: $y = 5 s i n x$

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

Example: $y = \frac{1}{2} s i n x$

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

Example: $y = - s i n x$

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

Example: $y = - 3 s i n x$

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: $y = s i n x + 2$

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

Example: $y = s i n x - 1$

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: $y = 2 s i n x + 3$

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

Example: $y = \frac{1}{2} s i n x - 1$

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

Example: $y = - 2 s i n x + 5$

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: $y = s i n 2 x$

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: $y = s i n \frac{1}{3} x$

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

Example: $y = s i n (- 4 x)$

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: $y = s i n (x - 60) °$

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

Example: $y = s i n (x + 45) °$

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: $y = s i n (2 x + 60) °$

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

Example: $y = s i n (\frac{1}{2} x - 30) °$

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: $y = 3 s i n (x + 25) °$

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: $y = - 2 s i n 3 x - 4$

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: $y = 5 s i n (2 x - 30) ° + 1$

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 $y = a s i n b x^{\circ}$ is shown in the diagram.

State the values of $a$ and $b .$

The basic $y = s i n x$ graph has amplitude 1 but this graph has amplitude 4, so $a = 4 .$

There are 3 cycles from 0° to 360°. So $b = 3 .$

Example 2 (non-calculator)

SQA National 5 Maths 2018 P1 Q6

Part of the graph of $y = a c o s b x^{\circ}$ is shown in the diagram.

State the values of $a$ and $b .$

The basic $y = c o s x$ graph has amplitude 1 but this graph has amplitude 5, so $a = 5 .$

The period of this graph is 90°, so there are $4$ cycles from 0° to 360°. So $b = 4 .$

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 $y = 3 c o s (x + 45)^{\circ}$ is shown in the diagram.

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

The graph of the basic $y = c o s x$ graph has a minimum turning point at (180,–1).

The equation tells us that this graph is stretched 3 times as tall and moved 45° left.

So A is (135, –3).

Example 4 (non-calculator)

SQA National 5 Maths 2021 P1 Q13

The graph of $y = a c o s x^{\circ} + b,$ $0 \leq x \leq 360,$ is given.

State the values of $a$ and $b .$

The given graph has a minimum value of 1 and a maximum value of 5, so its total height (from bottom to top) is 5 – 1 = 4. But the total height of the basic $y = c o s x$ graph is only 2 (from –1 up to 1) so the given graph is twice as tall, which tells us that $a = 2 .$

The graph of $y = 2 c o s x^{\circ}$ (without the ' $+ b$ ') has a minimum value of –2 and a maximum value of 2, so we can see that the given graph has been moved up 3. Therefore $b = 3 .$

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 $y = a s i n b x^{\circ}$ is shown in the diagram.

(a) State the value of $a .$
(b) State the value of $b .$

(a) This graph is 3 times as tall as the basic $y = s i n x$ graph, so $a = 3 .$

(b) The period of this graph is 45° so 8 cycles would fit from 0° to 360°. This tells us that $b = 8 .$

Example 6 (non-calculator)

SQA National 5 Maths 2023 P1 Q13

Part of the graph of $y = c o s (x + a)^{\circ} + b$ is shown.

(a) State the value of $a .$
(b) State the value of $b .$

(a) The ' $+ a$ ' inside the brackets represents a horizontal translation: left if $a > 0$ or right if $a < 0 .$

The minimum turning point on the basic $y = c o s x$ graph occurs when $x = 180^{\circ}$ but on the given graph we can see that it occurs when $x = 210^{\circ} .$ The given graph is therefore 30° to the right of the basic graph, so $a = - 30 .$

(b) The basic $y = c o s x$ graph has a minimum value of –1 and a maximum value of 1, so we can see that the given graph has been translated up 1, meaning that $b = 1 .$

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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

Click here to study the trig graphs notes on National5.com.

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National 5 MathsTrigonometric Graphs

Course content

Textbook page references

N5 Maths revision course

Basic trig graphs

Change of amplitude

Vertical translation

Vertical transformations

Multiple angle

Phase angle

Horizontal transformations

2D transformations

Example 1 (non-calculator)

Example 2 (non-calculator)

Recommended student book

Example 3 (non-calculator)

Example 4 (non-calculator)

N5 Maths practice papers

Example 5 (non-calculator)

Example 6 (non-calculator)

Need a Nat 5 Maths tutor?

Past paper questions

Other great resources

National 5 Maths
Trigonometric Graphs