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Describing Transformations of Radical Functions

Describing Transformations of Radical Functions 1.2 - 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}
Horizontal Stretch or Shrink Horizontal stretch, 0<b<1y=f(bx)\begin{gathered} \text{Horizontal stretch, } 0<\textcolor{magenta}{b}<1\\ y=f(\textcolor{magenta}{b}x) \end{gathered}
Horizontal shrink, b>1y=f(bx)\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 23.\frac{2}{3}. To perform this transformation, we will multiply the input by 1÷23=32.1\div \frac{2}{3}=\textcolor{magenta}{\frac{3}{2}}. Let hh be the function that represents the shrink of f.f. h(x)=f(32x)h(x)=(32)6x\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 4{\color{#009600}{4}} units. To do so, we will add 4{\color{#009600}{4}} to the input of hh to obtain g.g. g(x)=h(x+4)g(x)=(32)6(x+4)\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)=(32)6(x+4)g(x)=\sqrt{\left(\dfrac{3}{2}\right)6(x+4)}
g(x)=182(x+4)g(x)=\sqrt{\dfrac{18}{2}(x+4)}
g(x)=9(x+4)g(x)=\sqrt{9(x+4)}
g(x)=9x+36g(x)=\sqrt{9x+36}