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# Describing Transformations of Radical Functions

## Describing Transformations of Radical Functions 1.13 - Solution

We want to use the given graph to write a rule for $g.$ To do so, let's first identify the transformations that were applied to the parent function $f(x)=\sqrt[3]{x}.$

We can see that the graph of $g$ is a reflection in the $y\text{-}$axis followed by a horizontal translation right ${\color{#0000FF}{2}}$ units. Let's consider how these types of transformations affect the equation of the function.

Transformations of $f(x)$
Horizontal Translations $\begin{gathered} \text{Translation right } {\color{#0000FF}{h}} \text{ units, } {\color{#0000FF}{h}}>0 \\ y=f(x-{\color{#0000FF}{h}}) \end{gathered}$
$\begin{gathered} \text{Translation left } {\color{#0000FF}{h}} \text{ units, } {\color{#0000FF}{h}}>0 \\ y=f(x+{\color{#0000FF}{h}}) \end{gathered}$
Reflections $\begin{gathered} \text{In the } x\text{-axis}\\ y=\textcolor{deepskyblue}{\text{-}} f(x) \end{gathered}$
$\begin{gathered} \text{In the } y\text{-axis}\\ y=f(\textcolor{deepskyblue}{\text{-}} x) \end{gathered}$
Using the table, we can write the transformations of $f.$ Let's start by finding the rule for $h,$ whose graph is a reflection in the $y\text{-}$axis of the graph of $f.$ $\begin{gathered} h(x)=f(\textcolor{deepskyblue}{\text{-} }x) \quad \Leftrightarrow \quad h(x)=\sqrt[3]{\textcolor{deepskyblue}{\text{-} }x} \end{gathered}$ Finally, let's write the rule for $g,$ whose graph is a translation right $2$ units of the graph of $h.$ $\begin{gathered} g(x)=h(x-{\color{#0000FF}{2}}) \quad \Leftrightarrow \quad g(x)=\sqrt[3]{\text{-} (x-{\color{#0000FF}{2}})} \end{gathered}$ We can simplify the above formula by using the Distributive Property.
$g(x)=\sqrt[3]{\text{-} (x-2)}$
$g(x)=\sqrt[3]{\text{-} x+2}$
The desired function rule is $g(x)=\sqrt[3]{\text{-} x+2}.$