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# Describing Transformations of Radical Functions

## Describing Transformations of Radical Functions 1.2 - 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 $\begin{gathered} \text{Translation right } {\color{#009600}{h}} \text{ units, } {\color{#009600}{h}}>0 \\ y=f(x-{\color{#009600}{h}}) \end{gathered}$
$\begin{gathered} \text{Translation left } {\color{#009600}{h}} \text{ units, } {\color{#009600}{h}}>0 \\ y=f(x+{\color{#009600}{h}}) \end{gathered}$
Horizontal Stretch or Shrink $\begin{gathered} \text{Horizontal stretch, } 0<\textcolor{magenta}{b}<1\\ y=f(\textcolor{magenta}{b}x) \end{gathered}$
$\begin{gathered} \text{Horizontal shrink, } \textcolor{magenta}{b}>1 \\ y=f(\textcolor{magenta}{b}x) \end{gathered}$
The first transformation is a horizontal shrink by a factor of $\frac{2}{3}.$ To perform this transformation, we will multiply the input by $1\div \frac{2}{3}=\textcolor{magenta}{\frac{3}{2}}.$ Let $h$ be the function that represents the shrink of $f.$ $\begin{gathered} h(x)=f\left(\textcolor{magenta}{\dfrac{3}{2}}x\right) \quad \Leftrightarrow \quad h(x)=\sqrt{\left(\textcolor{magenta}{\dfrac{3}{2}}\right)6x} \end{gathered}$ The second transformation is a translation left ${\color{#009600}{4}}$ units. To do so, we will add ${\color{#009600}{4}}$ to the input of $h$ to obtain $g.$ $\begin{gathered} g(x)=h(x+{\color{#009600}{4}}) \\ \Updownarrow \\ g(x)=\sqrt{\left(\dfrac{3}{2}\right)6(x+{\color{#009600}{4}})} \end{gathered}$ Finally, we will simplify the above formula.
$g(x)=\sqrt{\left(\dfrac{3}{2}\right)6(x+4)}$
$g(x)=\sqrt{\dfrac{18}{2}(x+4)}$
$g(x)=\sqrt{9(x+4)}$
$g(x)=\sqrt{9x+36}$