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Trigonometry - Transformations

There are various transformations that can be applied to a parent trigonometric function in order to change its core attributes. A transformed function can be expressed algebraically as:

\(f(x) = a\sin(k(x-d)) + c\)

  • \(a\) represents the vertical stretch/compression factor and the function's amplitude
  • \(k\) represents the horizontal stretch/compression factor and affects the function's period and frequency
  • \(d\) represents the horizontal shift
  • \(c\) represents the vertical shift and the function's axis

Vertical Stretch/Compression

The value of \(a\) will determine whether the transformed function will be vertically stretched or compressed. This also affects the function's amplitude:

  • If \(a > 1\), the function will be vertically stretched by a factor of \(a\)
  • If \(0 < a < 1\), the function will be compressed by a factor of \(a\)

\(a > 0\)

Parent Sine Function graphed against vertically stretched Sine Function.

\(0 < a < 1\)

Parent Cosine Function graphed against vertically compressed Cosine Function.

Horizontal Stretch/Compression

The value of \(k\) will determine whether the transformed function will be horizontally stretched or compressed. This also affects the function's period and frequency:

  • If \(0 < k < 1\), the function will be horizontally stretched by a factor of \(1/k\)
  • If \(k > 1\), the function will be horizontally compressed by a factor of \(1/k\)
  • The period is related to the \(k\) value through the equation \(p = 2\pi/l\)

\(0 < k < 1\)

Parent Sine Function graphed against horizontally stretched Sine Function.

\(k > 1\)

Parent Sine Function graphed against horizontally compressed Sine Function.

Horizontal Shift

The value of \(d\) will determine whether the transformed function will be shifted left or right:

  • If \(d > 0\), the function will be shifted right
  • If \(d < 0\), the function will be shifted left

\(d > 0\)

Parent Cosine Function graphed against Cosine Function shifted to the right.

\(d < 0\)

Parent Cosine Function graphed against Cosine Function shifted to the left.

Vertical Shift

The value of \(c\) will determine whether the transformed function will be shifted upward or downward. This also affects the function's axis:

  • If \(c > 0\), the function will be shifted upward
  • If \(c < 0\), the function will be shifted downward

\(c > 0\)

Parent Sine Function graphed against Sine Function shifted upward.

\(c < 0\)

Parent Sine Function graphed against Sine Function shifted downward.

Reflections

  • If \(a < 0\), the function will either be vertically stretched or compressed with a reflection in the \(x\)-axis
  • If \(k < 0\), the function will either be horizontally stretched or compressed with a reflection in the \(y\)-axis

\(a < 0\)

Parent Tangent Function graphed against Tangent Function reflected in the x-axis.

\(k < 0\)

Parent Sine Function graphed against Sine Function reflected in the y-axis.

Enter in the values for the sinusoidal function or click on the button to generate random values. Entering these values will generate a graph comparing the base sinusoidal function to the transformed one.






Example

For the function \(f(x) = -3\sin(x - \pi)-1\):

  1. Identify the parent function
  2. Identify the transformations that were applied
  3. Identify the domain and range
  4. Sketch the transformed function over \(2\) cycles

i. We can identify the parent function as \(\boldsymbol{f(x) = \sin(x)}\).

ii. We can now determine the transformations as such:

  • \(\boldsymbol{a = -3}\); vertical stretch by a factor of \(3\) and reflection in the \(x\)-axis
  • \(\boldsymbol{d = \pi}\); shifted right by \(\pi\) units
  • \(\boldsymbol{c = -1}\); shifted downward \(1\) unit

iii. We can identify the domain and range as such:

  • Domain: \(\boldsymbol{\{x\in\mathbb{R}\}}\)
  • Range: \(\boldsymbol{\{y\in\mathbb{R} |-4 \leq y \leq 2\}}\)

iv. We can draw our transformed graph as such:

Graph of transformed sine function expressed as f(x)=-3sin(x-π)-1.

For the function \(g(x) = \cfrac{1}{5}\cos(4x + 2\pi)\):

  1. Identify the parent function
  2. Identify the asymptotes
  3. Identify the domain and range
  4. Sketch the transformed function over \(2\) cycles

i. We can identify the parent function as \(\boldsymbol{g(x) = \cos(x)}\).

ii. After shifting some of the values in the transformed equation, we can represent it as:

\(g(x) = \cfrac{1}{4}\cos\left[4\left(x + \cfrac{\pi}{2}\right)\right]\)

  • \(\boldsymbol{a = 1/5}\); vertical compression by a factor of \(1/5\)
  • \(\boldsymbol{k = 4}\); horizontal compression by a factor of \(1/4\)
  • \(\boldsymbol{d = -\pi/2}\); shifted left by \(\pi/2\) units

iii. We can identify the domain and range as such:

  • Domain: \(\boldsymbol{\{x\in\mathbb{R}\}}\)
  • Range: \(\boldsymbol{\{y\in\mathbb{R} |-0.2 \leq y \leq 0.2\}}\)

iv. We can draw our transformed graph as such:

Graph of transformed cosine function expressed as g(x)=1/5cos(4x-2π).
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