\begin{frame} \frametitle{Slant Asymptotes} \begin{exampleblock}{} Sketch the graph of $f(x) = \frac{x^3}{2x^2+1}$. \vspace{-1ex} \begin{overlayarea}{\textwidth}{2cm} \begin{itemize} \only<1-3> { \item $x$- and $y$-intercept: \quad $(0,0)$ } \only<2-3> { \item inflection points: $(-\sqrt{\frac{3}{2}},-\frac{3}{8}\sqrt{\frac{3}{2}})$, $(0,0)$ and $(\sqrt{\frac{3}{2}},\frac{3}{8}\sqrt{\frac{3}{2}})$ } \only<3-3> { \item slant asymptote: \quad $y = \frac{1}{2}x$ } \only<4-> { \item increasing on $(-\infty,\infty)$ and $f'(0) = 0$ \item concave up on $(-\infty,-\sqrt{3/2})$ and $(0,\sqrt{3/2})$ \item concave down on $(-\sqrt{3/2},0)$ and $(\sqrt{3/2},\infty)$ } \end{itemize} \end{overlayarea}\vspace{-1ex} \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=1cm,yscale=1.75] {\def\diabordery{.25cm} \diagram{-4}{4}{-2}{2}{1}} \diagramannotatez \diagramannotatex{-2,-1,1,2} \diagramannotatey{-1,1} \node[include=cred,minimum size=3mm] at (0,0) {}; \pause \node[include=cgreen] at ({-pow(3/2,.5)},{-3/8*pow(3/2,.5)}) {}; \node[include=cgreen] at (0,0) {}; \node[include=cgreen] at ({pow(3/2,.5)},{3/8*pow(3/2,.5)}) {}; \pause \draw[red,dashed] (-4,-2) -- (4,2); \pause\pause \begin{scope}[ultra thick] \draw[cgreen,ultra thick] plot[smooth,domain=-4:0,samples=200] function{x**3/(2*x**2+1)}; \pause \draw[cgreen,ultra thick] plot[smooth,domain=0:4,samples=200] function{x**3/(2*x**2+1)}; \end{scope} \end{tikzpicture} } \end{center} \end{exampleblock} \vspace{10cm} \end{frame}