5.2.2 Transformations of Trigonometric Functions

Transformations of trigonometric functions involve altering the basic sine, cosine, and tangent graphs to produce shifts, stretches, compressions, and reflections. These transformations allow us to model various periodic phenomena more accurately.

1. Vertical Stretching/Compressing:

This transformation changes the amplitude of the trigonometric function.

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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.

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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.

2. Horizontal Stretching/Compressing:

This transformation changes the period of the trigonometric function.

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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).

Period Calculation:

For $\ y = \sin(Bx) \ and \ y = \cos(Bx) :$ $\text{Period} = \frac{360^\circ}{|B|} \quad \text{or} \quad \frac{2\pi}{|B|}$
For $\ y = \tan(Bx) :$ $\text{Period} = \frac{180^\circ}{|B|} \quad \text{or} \quad \frac{\pi}{|B|}$

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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.

3. Vertical Shifting:

This transformation shifts the graph up or down along the y-axis.

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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.

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Example:

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

4. Horizontal Shifting:

This transformation shifts the graph left or right along the x-axis.

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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.

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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 $\ :highlight[\frac{\pi}{4} radians (or 45°) to the left]$ .

5. Reflections:

Reflections occur when the graph is flipped across one of the axes.

Reflection across the $x$ $x$ -axis:
- This occurs when $\ A$ is negative in $\ y = A\sin(x) \, \ y = A\cos(x) \, or \ y = A\tan(x) .$
- The graph is inverted, flipping all points across the x-axis.
- Example: $\ y = -\sin(x)$ flips the sine wave upside down.
Reflection across the $y$ $y$ -axis:
- This occurs when $B$ is negative in $\ y = \sin(Bx) \, \ y = \cos(Bx) \, or \ y = \tan(Bx) .$
- The graph is reflected horizontally.
- Example: $\ y = \sin(-x)$ produces the same graph as $\ y = \sin(x)$ , but the graph is flipped horizontally (which is identical for sine and cosine due to their symmetry, but not for tangent).

6. Combining Transformations:

Trigonometric functions can undergo multiple transformations simultaneously. For example, $\ y = 2\sin(3x - 45^\circ) + 1$ involves:

A vertical stretch by a factor of $2$ (amplitude is doubled).
A horizontal compression by a factor of $\ \frac{1}{3} \ (period is \ 120^\circ ).$
A phase shift to the right by $45°$ .
A vertical shift upwards by $1$ unit.

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Example:

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

Solution:

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

Summary:

Understanding and applying transformations of trigonometric functions allow us to manipulate and interpret these functions in various contexts. Each transformation—whether it's shifting, stretching, or reflecting—affects the graph in a specific way, and combining them enables the modelling of complex periodic behaviours.

Transformations of Trigonometric Functions Simplified Revision Notes for A-Level OCR Maths Pure

5.2.2 Transformations of Trigonometric Functions

1. Vertical Stretching/Compressing:

2. Horizontal Stretching/Compressing:

3. Vertical Shifting:

4. Horizontal Shifting:

5. Reflections:

6. Combining Transformations:

Example:

Summary:

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