Transformations of Trigonometric Functions (AQA A-Level Mathematics): Revision Notes

• General Form:$\ y = A \sin(x) \, \ y = A \cos(x) \, or \ y = A \tan(x)$
• Effect: The amplitude (height of the wave) is multiplied by the absolute value of $\ A$.
• If $\ |A| > 1$, the graph is stretched vertically.
• If $\ 0 < |A| < 1$, the graph is compressed vertically.
• If $\ A < 0$, the graph is also reflected across the $x$-axis.

Example:

• $\ y = 2\sin(x) :$ The amplitude doubles from $1$ to $2$.
• $\ y = \frac{1}{2}\cos(x)$: The amplitude is halved from $1$ to $0.5$.
• $\ y = -\tan(x)$: The graph of $\ \tan(x)$ is reflected across the $x$-axis.

• General Form: $\ y = \sin(Bx) \, \ y = \cos(Bx) \, or \ y = \tan(Bx)$
• Effect: The period of the function is affected by the value of $\ B .$
• If $\ |B| > 1$, the graph is compressed horizontally (the period decreases).
• If $\ 0 < |B| < 1 ,$ the graph is stretched horizontally (the period increases).

Example:

• $\ y = \sin(2x)$: The period is $\ :highlight[180^\circ] \ (or \ \pi \ radians),$ so the graph completes one cycle in half the usual time.
• $\ y = \cos\left(\frac{1}{2}x\right)$: The period is $\ :highlight[720^\circ] \ (or \ 4\pi \ radians)$, doubling the period.

• General Form: $\ y = \sin(x) + C \, \ y = \cos(x) + C \, or \ y = \tan(x) + C$
• Effect: The entire graph is shifted vertically by $\ C$ units.
• If $\ C > 0$, the graph shifts up.
• If $\ C < 0 ,$ the graph shifts down.

Example:

• $\ y = \sin(x) + 2$: The sine wave is shifted $:highlight[2 units upwards]$.
• $\ y = \cos(x) - 1$: The cosine wave is shifted $:highlight[1 unit downwards]$.

• General Form: $\ y = \sin(x - D) \, \ y = \cos(x - D) \, or \ y = \tan(x - D)$
• Effect: The entire graph is shifted horizontally by $\ D$ units.
• If $\ D > 0$, the graph shifts to the right.
• If $\ D < 0$, the graph shifts to the left.

Example:

• $\ y = \sin(x - 30^\circ) :$ The sine wave is shifted $:highlight[30° to the right]$.
• $\ y = \cos(x + \frac{\pi}{4})$: The cosine wave is shifted $\ :highlight[\frac{\pi}{4} radians (or 45°) to the left]$.

Example:

Problem: Sketch the graph of $\ y = -3\cos(2x + 60^\circ) - 2 .$

Solution:

1. Amplitude: The amplitude is $:highlight[3]$, but because of the negative sign, the graph is reflected across the x-axis.
2. Period: The period is $\ \frac{360^\circ}{2} = :highlight[180^\circ] .$
3. Phase Shift: The graph shifts to the left by $\ \frac{60^\circ}{2} = :highlight[30^\circ] .$
4. Vertical Shift: The graph is shifted down by 2 units.

Steps to Sketch:

• Start with the basic cosine curve.
• Reflect it across the $x$-axis.
• Compress it horizontally so it repeats every $180°$.
• Shift the graph $30°$ to the left.
• Finally, shift the entire graph down by $2$ units.