\begin{frame} \frametitle{Power Functions: Special Cases} \begin{exampleblock}{} We consider $x^n$ with $n$ a positive integer. \end{exampleblock} \pause \begin{itemize} \item For even $n$ the graph similar to the parabola $x^2$. \item For odd $n$ the graph looks similar to $x^3$. \end{itemize} \medskip \begin{minipage}{.49\textwidth} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=0ex] \diagram{-2}{2}{-2}{2}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{-1,1} \begin{scope}[ultra thick] \draw[cblue] plot[smooth,domain=-1.27:1.27,samples=20] (\x,{pow(\x,3)}); \draw[cred] plot[smooth,domain=-1.15:1.15,samples=20] (\x,{pow(\x,5)}); \draw[cgreen] plot[smooth,domain=-1.08:1.08,samples=20] (\x,{pow(\x,9)}); \draw[cblue] (-2.1cm,-2.8cm) -- node [at end,right] {$x^3$} +(.8cm,0); \draw[cred] (-.6cm,-2.8cm) -- node [at end,right] {$x^5$} +(.8cm,0); \draw[cgreen] (.9cm,-2.8cm) -- node [at end,right] {$x^9$} +(.8cm,0); \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} \begin{minipage}{.49\textwidth} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-2}{2}{-1}{3}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{-1,1} \begin{scope}[ultra thick] \draw[cblue] plot[smooth,domain=-1.75:1.75,samples=20] (\x,{pow(\x,2)}); \draw[cred] plot[smooth,domain=-1.32:1.32,samples=20] (\x,{pow(\x,4)}); \draw[cgreen] plot[smooth,domain=-1.2:1.2,samples=20] (\x,{pow(\x,6)}); \draw[cblue] (-2.1cm,-1.8cm) -- node [at end,right] {$x^2$} +(.8cm,0); \draw[cred] (-.6cm,-1.8cm) -- node [at end,right] {$x^4$} +(.8cm,0); \draw[cgreen] (.9cm,-1.8cm) -- node [at end,right] {$x^6$} +(.8cm,0); \node (a) [include=black,minimum size=1mm] at (1,1) {}; \node[r=(a)] {$(1,1)$}; \end{scope} \end{tikzpicture} } \end{minipage} \pause \begin{block}{} If $n$ increases, then the graph of $x^n$ becomes flatter near $0$, and steeper for $|x| \ge 1$. \end{block} \vspace{10cm} \end{frame}