### Basic functions: Power functions

### Transformations of power functions

Transformation of a power function By horizontal and vertical translation, and by vertical multiplication of the graph of the power function \(f(x)=x^p\) the graph of the function \[g(x)=a\cdot (x+c)^p + d\quad\text{with parameters }a,c,d\] can be created. For \(p=-1\) an arbitrary fractional linear function can be constructed and for \(p=2\) an arbitrary quadratic function can be constructed in this way.

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.