Basic functions: Power functions
Transformations of power functions
To illustrate transformations of power functions, we consider the horizontal and vertical translation as well as the vertical scaling of the graph of the power function \(f(x)=x^3\).
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Vertical translation
We translate the graph of \(f(x)=x^3\) vertically over a distance \(\blue{d}\).
The new function becomes \[g(x)=x^3+\blue{d}\] The point of symmetry of \(g\) is \((0,\blue{d})\), i.e., the graph of \(g\) does not change when reflected in \((0,\blue{d})\).
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Horizontal translation
We translate the graph of \(f(x)=x^3\) horizontally over a distance \(\blue{c}\).
The new function becomes \[g(x)=(x+\blue{c})^3\] The point of symmetry of \(g\) is \((-\blue{d},0)\), i.e., the graph of \(g\) does not change when reflected in \((-\blue{d},0)\).
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Vertical scaling
We scale the graph of \(f(x)=x^3\) vertically by \(\blue{a}\).
The new function becomes \[g(x)=\blue{a}\cdot x^3\] The symmetry point of the power function remains in place.
If \(a=-1\), then the graph of \(g\) is the mirror image of the graph of \(f\) in the horizontal axis, but also in the vertical axis.
