Exercise 1.3.31

Question

\begin{enumerate}[label=(\alph*)]
        \item How is the graph of $y = f (|x |)$ related to the graph of $f$?
        \item Sketch the graph of $y = \sin |x |.$
        \item Sketch the graph of $y =\sqrt{|x |}$.
    \end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item To obtain the graph of $y=f(|x|)$, we reflect about the $y$-axis the portion of the graph to the right of the $y$-axis.%The graph of $y=f(|x|)$ is obtained by reflecting the part of the graph of $f$ to the right of the $y$-axis about the $x$-axis.
    \item $$$$\begin{figure}[H]\centering\begin{tikzpicture}
    \begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}, width=6.5cm]
         \addplot[domain=-5:5, samples=50, mark=none, color=red, dashed, ultra thick] {sin(deg(x))};
         \legend{$\sin x$};
    \end{axis}    
    \end{tikzpicture} \qquad
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm},width=6.5cm]
         \addplot[domain=-5:5, samples=1000, mark=none, color=violet, ultra thick] {sin(abs(deg(x)))};
         \legend{$\sin(|x|)$};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}

\item $$$$\begin{figure}[H]\centering\begin{tikzpicture}
    \begin{axis}[axis lines=middle, unit vector ratio={1 1}, ymin=0, ymax=2.5, xmin=0, xmax=5, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}, width=0.3	extwidth]
         \addplot[domain=-1:5, samples=100, mark=none, color=red, dashed, ultra thick] {sqrt(x)};
         \legend{$\sqrt{x}$};
    \end{axis}    
    \end{tikzpicture} \quad
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, unit vector ratio={1 1}, ymin=0, ymax=2.5,  xmin=-5, xmax=5, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm},width=0.6	extwidth]
         \addplot[domain=-5:5, samples=1000, mark=none, color=violet, ultra thick] {sqrt(abs(x))};
         \legend{$\sqrt{|x|}$};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
\end{enumerate}

Exercise 1.3.31

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Exercise 1.3.31

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