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\title{Curve sketching}
\date{November 1, 2021}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\mtd{
To sketch $y=f(x)$:

\begin{enumerate}[1.]

\item {\bf Domain}. Where is the function defined?

\item {\bf Intercepts}.  $f(0)$ is the $y$-intercept. Find $x$-intercept(s) by solving $f(x) = 0$ for $x$.

\item {\bf Symmetry}.
\subitem {\bf Even} if $f(-x) = f(x)$ for all $x$ in the domain ($y$-axis symmetry).
\subitem {\bf Odd} if $f(-x) = -f(x)$ for all $x$ in the domain (origin symmetry).
\subitem {\bf Periodic} if it repeats (like $y = \sin x$ or other trig functions).

\item {\bf Asymptotes}.
\subitem {\bf Horizontal} (if any) at $\lim_{x\to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$.
\subitem {\bf Vertical} (if any) at places where $\lim_{x\to a^+} f(x) = \pm \infty$ or $\lim_{x\to a^-} f(x) = \pm \infty$ (watch for division by $0$, which you should already have done in step 1). Determine how the function approaches any asymptotes.

\item {\bf Increasing/Decreasing}. Use the first derivative.

\item {\bf Local Extremes}. Use the information from the previous step and find the extreme values.

\item {\bf Concavity and Inflection Points}. Use the second derivative.

\item {\bf Draw it}. Use as many steps above as you can to choose a scale for your axes and draw an accurate graph. You may also want to plot extra points to help you see how everything fits together.
\end{enumerate}
}

\prob{ 
Use the curve sketching guidelines above to draw the graph of $\displaystyle y=(x^2-1)^3$.

\vspace{.1in}

\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-5,-5) grid (5,5);
\end{tikzpicture}
}

\newpage

\prob{
Use the curve sketching guidelines to draw the graph of
$\displaystyle y=\frac{x}{x^3 -1}$.

\vspace{.1in}

\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-5,-5) grid (5,5);
\end{tikzpicture}
}

\end{document}
