Combinations Of Transformations
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Find the equation of the image of \(y = \sqrt x \), for a translation of 3 units in the positive direction of the \(x\)-axis followed by a dilation of a factor of 2 from the \(x\)-axis.
\begin{align}
&(x, y) \rightarrow(x+3, y) \rightarrow(x+3,2 y) \\
&x^{\prime}=x+3,\quad\; y^{\prime}=2 y \\
&x=x^{\prime}-3 \qquad y=\frac{1}{2} y^{\prime} \\
&\begin{aligned}
y=\sqrt{x} \rightarrow \; \frac{1}{2} y^{\prime}&=\sqrt{x^{\prime}-3} \\
y^{\prime}&=2 \sqrt{x^{\prime}-3} \\
g(x)&=2 \sqrt{x-3}
\end{aligned}
\end{align}
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