Exponentials - Transformations

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

\(f(x) = a(b)^{k(x-d)} + c\)

• \(a\) represents the vertical stretch/compression factor
• \(k\) represents the horizontal stretch/compression factor
• \(d\) represents the horizontal shift
• \(c\) represents the vertical shift and affects the function's Horizontal Asymptote

Vertical Stretch/Compression

The value of \(a\) will determine whether the transformed function will be vertically stretched or compressed:

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

Horizontal Stretch/Compression

The value of \(k\) will determine whether the transformed function will be vertically stretched or compressed:

• 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 \(1/k\)

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

Vertical Shift

The value of \(c\) will determine whether the transformed function will be shifted upward or downward.:

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

NOTE: The vertical shift will affect the function's Horizontal Asymptote.

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

Example

For the function \(u(x) = 0.6(3)^{4x-12}\):

1. Identify the transformations that were applied
2. Identify the Horizontal Asymptote
3. Determine the domain and range
4. Sketch the transformed function

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

\(u(x) = 0.6(2.5)^{4(x-3)}\)

We can determine the transformations as such:

• \(\boldsymbol{a = 0.6}\); vertical compression by a factor of \(0.6\)
• \(\boldsymbol{k = 4}\); horizontal compression by a factor of \(0.25\)
• \(\boldsymbol{d = 3}\); shifted right by \(3\) units

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Since \(c = 0\), the Horizontal Asymptote remains at \(\boldsymbol{y = 0}\).

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

• Domain: \(\{x\in\mathbb{R}\}\)
• Range: \(\{y\in\mathbb{R} | y > 0)\}\)

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iv. We can draw our transformed graph as such:

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