\begin{frame} \frametitle{Exponential Functions: Examples} \begin{center} \begin{tikzpicture}[default,baseline=0cm,scale=1.2,nodes={scale=.9}] {\def\diaborderx{.7cm} \def\diabordery{.6cm} \diagram{-2.5}{2.5}{-.5}{3}{1}} \diagramannotatez \diagramannotatexx{-1,1} \diagramannotatey{} \begin{scope}[ultra thick] \draw[orange] plot[smooth,domain=-2.5:2.5,samples=300] (\x,{1^\x}) node [right] {$1^x$}; \draw[cblue] plot[smooth,domain=-1.6:2.5,samples=300] (\x,{.5^\x}); \node[cblue,above] at (-1.6,{.5^(-1.6)}) {$\left(\frac{1}{2}\right)^x$}; \draw[cgreen] plot[smooth,domain=-2.5:1.6,samples=300] (\x,{2^\x}); \node[cgreen,above] at (1.6,{2^(1.6)}) {$2^x$}; \draw[gray] plot[smooth,domain=-2.5:2.5,samples=300] (\x,{1.5^\x}); \node[gray,right] at (2.5,{1.5^(2.5)}) {$1.5^x$}; \draw[pink!80!red] plot[smooth,domain=-0.8:2.5,samples=300] (\x,{.25^\x}); \node[pink!80!red,above] at (-0.8,{.25^(-0.8)}) {$\left(\frac{1}{4}\right)^x$}; \draw[cred] plot[smooth,domain=-2.5:0.8,samples=300] (\x,{4^\x}); \node[cred,above] at (0.8,{4^(0.8)}) {$4^x$}; \end{scope} \end{tikzpicture} \end{center} Properties: \begin{itemize} \item All exponential functions pass through $(0,1)$ \textcolor{gray}{(since $a^0 = 1$)} \item Larger base $a$ yields more rapid growth for $x > 0$. \end{itemize} \end{frame}