The resulting function of y=e^(−3x), when vertically stretched by a factor of 4, reflected across the y-axis, and then shifted up 5 units is y=[4e^(−3x)] + 5.
How does the transformation of a function happen?
The transformation of a function may involve any change.
Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs) etc.
If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:
Horizontal shift (also called phase shift):
- Left shift by c units, y=f(x+c) (same output, but c units earlier)
- Right shift by c units, y=f(x-c)(same output, but c units late)
Vertical shift
- Up by d units:
- Down by d units: y = f(x) - d
Stretching:
- Vertical stretch by a factor k: [tex]y = k \times f(x)[/tex]
- Horizontal stretch by a factor k: [tex]y = f(\dfrac{x}{k})[/tex]
Given the function [tex]y=e^{-3x}[/tex] is vertically stretched by a factor of 4. Therefore, the function will be written as,
[tex]y = 4 \times e^{-3x}[/tex]
Now, the function is shifted up by 5 units, therefore, the function will become,
[tex]y = (4e^{-3x})+5[/tex]
Hence, the resulting function of y=e^(−3x) , when vertically stretched by a factor of 4, reflected across the y-axis, and then shifted up 5 units is y=[4e^(−3x)] + 5.
Learn more about Transforming functions:
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