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Graph the function \(\displaystyle f(x)=\begin{cases} \frac{4 x}{5} + 8,& x \leq -2\\ 9 - \frac{\left(x - 5\right)^{2}}{5},& -2 < x < 5\\ \frac{3 x}{5} + 3,& x \geq 5 \\ \end{cases}\)

The graph has three parts. It is a line on \(\displaystyle (-\infty, -2]\), a parabola on \(\displaystyle (-2,5)\), and a line on \(\displaystyle [5, \infty)\). Graphing each part gives:

Download Tikz \( \LaTeX \) 

\begin{tikzpicture}[font=\Large, scale=.4]
	\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-10,-10) grid (10,10);
	\draw[latex-latex, very thick] (-10.4,0)--(10.4,0) node[right]{$x$};
	\draw[latex-latex, very thick] (0,-10.4)--(0,10.4) node[above]{$y$};
	\foreach \x in { -10,-8,...,10 }{
		\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
		\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
		}
	\foreach \y in { -10,-8,...,10 }{
		\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
		\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
		}
	\draw[thick, color=blue, domain={-10.0}:{-2.0}, latex-, samples =350] plot(\x, {4*(\x)/5 + 8});
	\draw[thick, color=blue, domain={-2.0}:{5.0}, samples =350] plot(\x, {9 - ((\x) - 5)^2/5});
	\draw[thick, color=blue, domain={5.0}:{10.0}, -latex, samples =350] plot(\x, {3*(\x)/5 + 3});
		\draw[fill=black] (-2,32/5) circle (5pt);
		\draw[fill=white] (-2,-4/5) circle (5pt);
		\draw[fill=white] (5,9) circle (5pt);
		\draw[fill=black] (5,6) circle (5pt);
\end{tikzpicture}

Download \(\LaTeX\) 

\begin{question}Graph the function $f(x)=\begin{cases} \frac{4 x}{5} + 8,& x \leq -2\\ 9 - \frac{\left(x - 5\right)^{2}}{5},& -2 < x < 5\\ \frac{3 x}{5} + 3,& x \geq 5 \\ \end{cases}$\newline
\begin{tikzpicture}[font=\Large, scale=.4]
	\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-10,-10) grid (10,10);
	\draw[latex-latex, very thick] (-10.4,0)--(10.4,0) node[right]{$x$};
	\draw[latex-latex, very thick] (0,-10.4)--(0,10.4) node[above]{$y$};
	\foreach \x in { -10,-8,...,10 }{
		\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
		\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
		}
	\foreach \y in { -10,-8,...,10 }{
		\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
		\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
		}
\end{tikzpicture}

    \soln{0cm}{The graph has three parts. It is a line on $(-\infty, -2]$, a parabola on $(-2,5)$, and a line on $[5, \infty)$.  Graphing each part gives:\newline
\begin{tikzpicture}[font=\Large, scale=.4]
	\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-10,-10) grid (10,10);
	\draw[latex-latex, very thick] (-10.4,0)--(10.4,0) node[right]{$x$};
	\draw[latex-latex, very thick] (0,-10.4)--(0,10.4) node[above]{$y$};
	\foreach \x in { -10,-8,...,10 }{
		\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
		\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
		}
	\foreach \y in { -10,-8,...,10 }{
		\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
		\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
		}
	\draw[thick, color=blue, domain={-10.0}:{-2.0}, latex-, samples =350] plot(\x, {4*(\x)/5 + 8});
	\draw[thick, color=blue, domain={-2.0}:{5.0}, samples =350] plot(\x, {9 - ((\x) - 5)^2/5});
	\draw[thick, color=blue, domain={5.0}:{10.0}, -latex, samples =350] plot(\x, {3*(\x)/5 + 3});
		\draw[fill=black] (-2,32/5) circle (5pt);
		\draw[fill=white] (-2,-4/5) circle (5pt);
		\draw[fill=white] (5,9) circle (5pt);
		\draw[fill=black] (5,6) circle (5pt);
\end{tikzpicture}
}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
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\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
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\fi%
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\newcommand{\hideifShowSolution}[1]{%
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\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}
HTML for Canvas 
<p> <p>Graph the function  <img class="equation_image" title=" \displaystyle f(x)=\begin{cases} \frac{4 x}{5} + 8,& x \leq -2\\ 9 - \frac{\left(x - 5\right)^{2}}{5},& -2 < x < 5\\ \frac{3 x}{5} + 3,& x \geq 5 \\ \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D%5Cbegin%7Bcases%7D%20%5Cfrac%7B4%20x%7D%7B5%7D%20%2B%208%2C%26%20x%20%5Cleq%20-2%5C%5C%209%20-%20%5Cfrac%7B%5Cleft%28x%20-%205%5Cright%29%5E%7B2%7D%7D%7B5%7D%2C%26%20-2%20%3C%20x%20%3C%205%5C%5C%20%5Cfrac%7B3%20x%7D%7B5%7D%20%2B%203%2C%26%20x%20%5Cgeq%205%20%5C%5C%20%5Cend%7Bcases%7D%20" alt="LaTeX:  \displaystyle f(x)=\begin{cases} \frac{4 x}{5} + 8,& x \leq -2\\ 9 - \frac{\left(x - 5\right)^{2}}{5},& -2 < x < 5\\ \frac{3 x}{5} + 3,& x \geq 5 \\ \end{cases} " data-equation-content=" \displaystyle f(x)=\begin{cases} \frac{4 x}{5} + 8,& x \leq -2\\ 9 - \frac{\left(x - 5\right)^{2}}{5},& -2 < x < 5\\ \frac{3 x}{5} + 3,& x \geq 5 \\ \end{cases} " /> <br>
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HTML for Canvas
<p> <p>The graph has three parts. It is a line on  <img class="equation_image" title=" \displaystyle (-\infty, -2] " src="/equation_images/%20%5Cdisplaystyle%20%28-%5Cinfty%2C%20-2%5D%20" alt="LaTeX:  \displaystyle (-\infty, -2] " data-equation-content=" \displaystyle (-\infty, -2] " /> , a parabola on  <img class="equation_image" title=" \displaystyle (-2,5) " src="/equation_images/%20%5Cdisplaystyle%20%28-2%2C5%29%20" alt="LaTeX:  \displaystyle (-2,5) " data-equation-content=" \displaystyle (-2,5) " /> , and a line on  <img class="equation_image" title=" \displaystyle [5, \infty) " src="/equation_images/%20%5Cdisplaystyle%20%5B5%2C%20%5Cinfty%29%20" alt="LaTeX:  \displaystyle [5, \infty) " data-equation-content=" \displaystyle [5, \infty) " /> .  Graphing each part gives:<br>
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