9.6PureStretch

Transforming Trigonometric Graphs

The graph transformations you learned in Chapter 4 apply directly to trig graphs. Translations, stretches and reflections of $sin$ , $cos$ and $tan$ change the amplitude, period and position of the wave in predictable ways.

25 min Video by Zeeshan Zamurred Trigonometric Ratios

Edexcel AS Level Maths: 9.6 Transforming Trigonometric GraphsWatch the full walkthrough before the notes below.

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What you'll be able to do

Apply translations to trig graphs
Apply vertical and horizontal stretches
Interpret amplitude and period changes
Combine transformations of trig functions

Same rules as before

Everything carries over from general graph transformations: changes outside the function move/scale vertically (in the natural direction), changes inside act horizontally (opposite/reciprocal).

y = sin x + a : up a y = sin (x + a) : left a

Translations.

y = a sin x : amplitude a y = sin (b x) : period \div b

Stretches.

Amplitude and period

For $y = a sin (b x)$ , the $amplitude$ is $a$ (the wave now runs from $- a$ to $a$ ) and the $period$ becomes $\frac{36 0 ^{\circ}}{b}$ — a bigger $b$ squashes more waves into the same width.

y = a sin (b x) : amplitude = a, period = \frac{36 0 ^{\circ}}{b}

For cosine likewise; for tangent the base period is

18 0^{\circ}

1Amplitude is the outside factor:

3

2Period is

\frac{36 0 ^{\circ}}{2} = 18 0^{\circ}

Answeramplitude

3

, period

18 0^{\circ}

Tip — Outside factor → amplitude; inside factor → divides the period. Don’t swap them.

Reflections and combinations

A negative outside factor reflects in the $x$ -axis ( $y = - sin x$ ); combinations such as $y = 2 cos x - 1$ stack a stretch and a translation. Apply each transformation in turn and track the new max/min.

1Amplitude

2

(runs from

- 2

2

2Then shift down

1

(now from

- 3

1

Answeramplitude 2, shifted down 1; range

- 3 \leq y \leq 1

Formula recap

y = a sin (b x) + c

Amplitude a, period 360°/b, shift c.

period = \frac{36 0 ^{\circ}}{b}

Inside factor squashes the wave.

y = - sin x

Reflection in the x-axis.

Common mistakes to avoid

Saying y = sin(2x) has period 720°.

Inside factor DIVIDES the period: 360°/2 = 180°.

Treating the outside factor as a period change.

The outside factor sets the amplitude, not the period.

Key takeaways

Trig graphs transform with the same rules as any function.
y = a sin(bx): amplitude a, period 360°/b.
Outside factor = amplitude; inside factor divides the period.
Negative outside factor reflects in the x-axis; +c shifts vertically.

Test yourself

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