Exercise 1.3.6 (6-7)

Question

The graph of $y =\sqrt{3x- x^2}$ is given. Use transformations to create a function whose graph is as shown.

Khagan Gulusoy · Accepted Answer

\begin{enumerate}
    \item [6.] To get the graph in (6), we need to move the original graph $2$ units to the right (i.e., subtract two from $x$), then stretch it vertically by a factor of $2$ (i.e., multiply $f(x-2)$ by $2$). Thus, the function corresponding to the graph is:
    $$\forall x\in [2,5]:\quad g(x):=2f(x-2).$$
    In other words,
    $$\forall x\in [2,5]:\quad g(x)=2\sqrt{3(x-2)-\left(x-2\right)^{2}}.$$
    \item [7.] To get the graph in (7), we need to reflect the original graph about the $x$-axis (i.e., multiply $f(x)$ by $-1$). After that, we have to move it $4$ units to the left (i.e., add $4$ to $x$). Then, we need to shift the graph $1$ downward (i.e., subtract $1$ from $-f(x+4).$). Thus, the function corresponding to the graph is
    $$\forall x\in [-4,-1]:\quad h(x)=-f(x+4)-1.$$
    In other words,
    $$\forall x\in [-4,-1]:\quad h(x)=-\sqrt{3x+12-\left(x+4\right)^{2}}-1.$$
\end{enumerate}

Exercise 1.3.6 (6-7)

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Exercise 1.3.6 (6-7)

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