\begin{frame} \frametitle{Rational Functions} \vspace{-1ex} \begin{block}{} A \emph{rational function} $f$ is ratio of two polynomials: \begin{talign} f(x) = \frac{P(x)}{Q(x)} &&\text{where $P$ and $Q$ are polynomials} \end{talign} \end{block} \medskip\pause \begin{itemize} \item the domain of $\frac{P(x)}{Q(x)}$ is $\{\,x \mid Q(x) \ne 0\,\}$ \end{itemize} \pause \begin{center} \begin{minipage}{.4\textwidth} \scalebox{.6}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-3}{3}{-3}{3}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{-1,1} \begin{scope}[ultra thick] \draw[cblue] plot[smooth,domain=0.33:3,samples=100] (\x,{1/\x}); \draw[cblue] plot[smooth,domain=-3:-0.33,samples=100] (\x,{1/\x}); \end{scope} \end{tikzpicture} }\\\smallskip \centerline{$f(x) = \frac{1}{x}$} \end{minipage}\hspace{1cm} \begin{minipage}{.4\textwidth} \scalebox{.465}{ \begin{tikzpicture}[default,baseline=0ex] \diagram{-4}{4}{-3}{5}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{-1,1} \draw[cblue,ultra thick] plot[smooth,domain=-1.83:1.83,samples=20] (\x,{(2*pow(\x,4)-pow(\x,2)+1) / (10*pow(\x,2) - 40))}); \draw[cblue,ultra thick] plot[smooth,domain=2.21:3.95,samples=20] (\x,{(2*pow(\x,4)-pow(\x,2)+1) / (10*pow(\x,2) - 40))}); \draw[cblue,ultra thick] plot[smooth,domain=-3.95:-2.21,samples=20] (\x,{(2*pow(\x,4)-pow(\x,2)+1) / (10*pow(\x,2) - 40))}); \end{tikzpicture} }\\\smallskip \centerline{$f(x) = \frac{2x^4 - x^2 +1}{10x^2 - 40}$} \end{minipage} \end{center} \end{frame}