Exercise 1.3.4

Question

The graph of $f$ is given. Draw the graphs of the following functions.
    \begin{enumerate}[label=(\alph*)]
        \item $y=f(x)-3;$ 
        \item $y=f(x+1);$
        \item $y=1/2 \ f(x);$
        \item $y=-f(x).$
    \end{enumerate}
%    \begin{figure}[H] \centering
%    \begin{tikzpicture}
%    \begin{axis}[axis lines=middle, xmin=-3, xmax=4, ymax=4, ymin=-2, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
%         \addplot[domain=-2:0, samples=2, mark=none, color=red, ultra thick] {-1*x};
%         \addplot[domain=0:1, samples=2, mark=none, color=red, ultra thick] {3*x};
%         \addplot[domain=1:3, samples=2, mark=none, color=red, ultra thick] {3.5-0.5*x};
%         \addplot[mark=*, mark options={scale=1, fill=red}](-2,2);
%         \addplot[mark=*, mark options={scale=1, fill=red}](3,2);
%    \end{axis}    
%    \end{tikzpicture}
%    \end{figure}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item To draw $f(x)-3,$ we must shift the graph of $f$ 3 points downward:
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xmin=-3, xmax=4, ymax=4, ymin=-4, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-2:0, samples=2, mark=none, color=red, dashed, ultra thick] {-1*x};
         \addplot[domain=0:1, samples=2, mark=none, color=red, dashed, ultra thick] {3*x};
         \addplot[domain=1:3, samples=2, mark=none, color=red, dashed, ultra thick] {3.5-0.5*x};
         \addplot[domain=-2:0, samples=2, mark=none, color=red, violet, ultra thick] {-1*x-3};
         \addplot[domain=0:1, samples=2, mark=none, color=red, violet, ultra thick] {3*x-3};
         \addplot[domain=1:3, samples=2, mark=none, color=red, violet, ultra thick] {0.5-0.5*x};
         \addplot[mark=*, mark options={scale=1, fill=violet}](-2,-1);
         \addplot[mark=*, mark options={scale=1, fill=violet}](3,-1);
         \addplot[mark=*, mark options={scale=1, fill=red}](-2,2);
         \addplot[mark=*, mark options={scale=1, fill=red}](3,2);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    \item Since for all $x\in \mathbb{R}$ it must hold that 
    $$x+1\in [-2,3]$$
    (otherwise $f(x+1)$ is undefined), we must have $x\in [-3,2].$ To draw $f(x+1)$, we must shift $f$ 1 unit to the left. Thus, we get:
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xmin=-3, xmax=4, ymax=4, ymin=-1, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-2:0, samples=2, mark=none, color=red, dashed, ultra thick] {-1*x};
         \addplot[domain=0:1, samples=2, mark=none, color=red, dashed, ultra thick] {3*x};
         \addplot[domain=1:3, samples=2, mark=none, color=red, dashed, ultra thick] {3.5-0.5*x};
         \addplot[domain=-3:-1, samples=2, mark=none, color=red, violet, ultra thick] {-1*x-1};
         \addplot[domain=-1:0, samples=2, mark=none, color=red, violet, ultra thick] {3*x+3};
         \addplot[domain=0:2, samples=2, mark=none, color=red, violet, ultra thick] {3-0.5*x};
         \addplot[mark=*, mark options={scale=1, fill=violet}](-3,2);
         \addplot[mark=*, mark options={scale=1, fill=violet}](2,2);
         \addplot[mark=*, mark options={scale=1, fill=red}](-2,2);
         \addplot[mark=*, mark options={scale=1, fill=red}](3,2);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    \item To draw $1/2 \ f(x)$, we have to shrink every single part of the graph of $f$ vertically by a factor of $2:$
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xmin=-3, xmax=4, ymax=4, ymin=-1, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-2:0, samples=2, mark=none, color=red, dashed, ultra thick] {-1*x};
         \addplot[domain=0:1, samples=2, mark=none, color=red, dashed, ultra thick] {3*x};
         \addplot[domain=1:3, samples=2, mark=none, color=red, dashed, ultra thick] {3.5-0.5*x};
         \addplot[domain=-2:0, samples=2, mark=none, color=red, violet, ultra thick] {-0.5*x};
         \addplot[domain=0:1, samples=2, mark=none, color=red, violet, ultra thick] {1.5*x};
         \addplot[domain=1:3, samples=2, mark=none, color=red, violet, ultra thick] {1.75-0.25*x};
         \addplot[mark=*, mark options={scale=1, fill=violet}](-2,1);
         \addplot[mark=*, mark options={scale=1, fill=violet}](3,1);
         \addplot[mark=*, mark options={scale=1, fill=red}](-2,2);
         \addplot[mark=*, mark options={scale=1, fill=red}](3,2);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    \item We have to reflect the graph of $f$ about the $x$-axis:
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xmin=-3, xmax=4, ymax=4, ymin=-4, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-2:0, samples=2, mark=none, color=red, dashed, ultra thick] {-1*x};
         \addplot[domain=0:1, samples=2, mark=none, color=red, dashed, ultra thick] {3*x};
         \addplot[domain=1:3, samples=2, mark=none, color=red, dashed, ultra thick] {3.5-0.5*x};
         \addplot[domain=-2:0, samples=2, mark=none, color=red, violet, ultra thick] {x};
         \addplot[domain=0:1, samples=2, mark=none, color=red, violet, ultra thick] {-3*x};
         \addplot[domain=1:3, samples=2, mark=none, color=red, violet, ultra thick] {-3.5+0.5*x};
         \addplot[mark=*, mark options={scale=1, fill=violet}](-2,-2);
         \addplot[mark=*, mark options={scale=1, fill=violet}](3,-2);
         \addplot[mark=*, mark options={scale=1, fill=red}](-2,2);
         \addplot[mark=*, mark options={scale=1, fill=red}](3,2);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
\end{enumerate}

Exercise 1.3.4

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Exercise 1.3.4

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