\begin{frame} \frametitle{Power Functions: Special Cases} \begin{exampleblock}{} The power function $f(x) = x^{-1} = \frac{1}{x}$ is the \emph{reciprocal function}. \end{exampleblock} \begin{center} \begin{minipage}{.49\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}); \node (a) [include=black,minimum size=1mm] at (1,1) {}; \node[r=(a)] {$(1,1)$}; \node (b) [include=black,minimum size=1mm] at (-1,-1) {}; \node[l=(b)] {$(-1,-1)$}; \end{scope} \end{tikzpicture} } \end{minipage} \onslide<3->{ \begin{minipage}{.40\textwidth} \scalebox{.6}{ \begin{tikzpicture}[default,baseline=1cm] {\def\diax{P} \def\diay{V} \diagram{-1}{5}{-1}{5}{1}} \diagramannotatez \begin{scope}[ultra thick] \draw[cblue] plot[smooth,domain=0.4:5,samples=100] (\x,{2/\x}); \end{scope} \end{tikzpicture} } \end{minipage} } \end{center} \pause\smallskip \begin{exampleblock}{} This function arises in physics and chemistry. E.g. Boyle's law says that, when the temperature is constant, then the volume $V$ of a gas is inversely proportional to the pressure $P$: \begin{malign} V = \frac{C}{P} &&\text{where $C$ is a constant} \end{malign} \end{exampleblock} \vspace{10cm} \end{frame}