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Describing Transformations of Radical Functions

Describing Transformations of Radical Functions 1.13 - Solution

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We want to use the given graph to write a rule for g.g. To do so, let's first identify the transformations that were applied to the parent function f(x)=x3.f(x)=\sqrt[3]{x}.

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

Transformations of f(x)f(x)
Horizontal Translations Translation right h units, h>0y=f(xh)\begin{gathered} \text{Translation right } {\color{#0000FF}{h}} \text{ units, } {\color{#0000FF}{h}}>0 \\ y=f(x-{\color{#0000FF}{h}}) \end{gathered}
Translation left h units, h>0y=f(x+h)\begin{gathered} \text{Translation left } {\color{#0000FF}{h}} \text{ units, } {\color{#0000FF}{h}}>0 \\ y=f(x+{\color{#0000FF}{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}
Using the table, we can write the transformations of f.f. Let's start by finding the rule for h,h, whose graph is a reflection in the y-y\text{-}axis of the graph of f.f. h(x)=f(-x)h(x)=-x3\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,g, whose graph is a translation right 22 units of the graph of h.h. g(x)=h(x2)g(x)=-(x2)3\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)=-(x2)3g(x)=\sqrt[3]{\text{-} (x-2)}
g(x)=-x+23g(x)=\sqrt[3]{\text{-} x+2}
The desired function rule is g(x)=-x+23.g(x)=\sqrt[3]{\text{-} x+2}.