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# Describing Transformations of Radical Functions

## Describing Transformations of Radical Functions 1.14 - Solution

We want to write a rule for $g$ that represents the indicated transformations of the graph of $f.$ To do so, we will look at how the indicated transformations will affect the parent function. Then we can apply them.

Transformations of $f(x)$
Horizontal Translations $\begin{gathered} \text{Translation right } {\color{#009600}{h}} \text{ units, } {\color{#009600}{h}}>0 \\ y=f(x-{\color{#009600}{h}}) \end{gathered}$
$\begin{gathered} \text{Translation left } {\color{#009600}{h}} \text{ units, } {\color{#009600}{h}}>0 \\ y=f(x+{\color{#009600}{h}}) \end{gathered}$
Reflections $\begin{gathered} \text{In the } x\text{-axis}\\ y=\textcolor{deepskyblue}{\text{-}} f(x) \end{gathered}$
$\begin{gathered} \text{In the } y\text{-axis}\\ y=f(\textcolor{deepskyblue}{\text{-}} x) \end{gathered}$
The transformations given in the exercise are a reflection in the $y\text{-}$axis followed by a horizontal translation right ${\color{#009600}{1}}$ unit. We will first apply the reflection. Let $h$ be the function that represents the reflection of $f$ in the $y\text{-}$axis. Recalling that $f(x)=2\sqrt[3]{x-1},$ we can write its rule. $\begin{gathered} h(x)=f(\textcolor{deepskyblue}{\text{-}}x) \quad \Leftrightarrow \quad h(x)=2\sqrt[3]{\textcolor{deepskyblue}{\text{-}}x-1} \end{gathered}$ Next, we will apply the translation to $h$ to obtain $g.$ To do so, we will subtract ${\color{#009600}{1}}$ from the input of $h.$ $\begin{gathered} g(x)=h(x-{\color{#009600}{1}}) \\ \Updownarrow \\ g(x)=2\sqrt[3]{\text{-} (x-{\color{#009600}{1}})-1} \end{gathered}$ Finally, we will simplify the above formula.
$g(x)=2\sqrt[3]{\text{-} (x-1)-1}$
$g(x)=2\sqrt[3]{\text{-} x+1-1}$
$g(x)=2\sqrt[3]{\text{-} x}$