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Describing Transformations of Radical Functions

Describing Transformations of Radical Functions 1.14 - Solution

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We want to write a rule for gg that represents the indicated transformations of the graph of f.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)f(x)
Horizontal Translations Translation right h units, h>0y=f(xh)\begin{gathered} \text{Translation right } {\color{#009600}{h}} \text{ units, } {\color{#009600}{h}}>0 \\ y=f(x-{\color{#009600}{h}}) \end{gathered}
Translation left h units, h>0y=f(x+h)\begin{gathered} \text{Translation left } {\color{#009600}{h}} \text{ units, } {\color{#009600}{h}}>0 \\ y=f(x+{\color{#009600}{h}}) \end{gathered}
Reflections In the x-axisy=-f(x)\begin{gathered} \text{In the } x\text{-axis}\\ y=\textcolor{deepskyblue}{\text{-}} f(x) \end{gathered}
In the y-axisy=f(-x)\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-y\text{-}axis followed by a horizontal translation right 1{\color{#009600}{1}} unit. We will first apply the reflection. Let hh be the function that represents the reflection of ff in the y-y\text{-}axis. Recalling that f(x)=2x13,f(x)=2\sqrt[3]{x-1}, we can write its rule. h(x)=f(-x)h(x)=2-x13\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 hh to obtain g.g. To do so, we will subtract 1{\color{#009600}{1}} from the input of h.h. g(x)=h(x1)g(x)=2-(x1)13\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-(x1)13g(x)=2\sqrt[3]{\text{-} (x-1)-1}
g(x)=2-x+113g(x)=2\sqrt[3]{\text{-} x+1-1}
g(x)=2-x3g(x)=2\sqrt[3]{\text{-} x}