Exercise 1.4.9 (9-14)

Question

Make a rough sketch by hand of the graph of the function. Use the graphs given in Figures 3 and 15 and, if necessary, the transformations of Section 1.3.

\begin{enumerate}
  \setcounter{enumi}{8}
  \item $g(x)=3^{x}+1$

\item $h(x)=2\left(\frac{1}{2}\right)^{x}-3$

\item $y=-e^{-x}$

\item $y=4^{x+2}$

\item $y=1-\frac{1}{2} e^{-x}$

\item $y=e^{|x|}$ 
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}
\setcounter{enumi}{8}
\item We first draw the graph of $y=3^x$ and then shift it up by one unit and obtain the desired graph:
\begin{figure}[H]\centering
\begin{tikzpicture}
\begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, domain=-2:2, samples=200,]
\addplot[mark=none, color=blue, ultra thick,dashed] {3^x};
\addlegendentry{$y=3^x$};
\addplot[mark=none,color=red,ultra thick] {3^x+1};
\addlegendentry{$y=3^x+1$};
\end{axis}
\end{tikzpicture}
\end{figure}
\item We first draw the graph of $y=\left(\dfrac{1}{2}\right)^x$, then stretch it vertically by a factor of 2, shift it downwards by 3 units, and obtain the desired graph:
\begin{figure}[H]\centering
\begin{tikzpicture}
\begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, domain=-2:2, samples=200]
\addplot[mark=none, color=blue, ultra thick,dashed] {0.5^x};
\addlegendentry{$y=\left(\frac{1}{2}\right)^x$};
\addplot[mark=none,color=green!60!black,ultra thick,dashed] {2*0.5^x};
\addlegendentry{$y=2\left(\frac{1}{2}\right)^x$};
\addplot[mark=none,color=red,ultra thick] {2*0.5^x-3};
\addlegendentry{$y=2\left(\frac{1}{2}\right)^x-3$};
\end{axis}
\end{tikzpicture}
\end{figure}

\item To obtain the desired graph, we should reflect the graph of $y=e^x$ about the $y$-axis, and after that, about the $x$-axis:
\begin{figure}[H]\centering
\begin{tikzpicture}
\begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, domain=-2:2, samples=200,]
\addplot[mark=none, color=blue, ultra thick,dashed] {exp(x)};
\addlegendentry{$y=e^x$};
\addplot[mark=none,color=green!60!black,ultra thick,dashed] {exp(-x)};
\addlegendentry{$y=e^{-x}$};
\addplot[mark=none,color=red,ultra thick] {-exp(-x)};
\addlegendentry{$y=-e^{-x}$};
\end{axis}
\end{tikzpicture}
\end{figure}

\item To obtain the desired graph, we should shift the graph of $y=4^x$ by $2$ units to the left:
\begin{figure}[H]\centering
\begin{tikzpicture}
\begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, domain=-3:2, samples=200,ymax=10,xmin=-3]
\addplot[mark=none, color=blue, ultra thick,dashed] {4^x};
\addlegendentry{$y=4^x$};
\addplot[mark=none,color=red,ultra thick] {4^(x+2)};
\addlegendentry{$y=4^{x+2}$};
\addplot[mark=*] table {
    0 1
    -2 1
};
\end{axis}
\end{tikzpicture}
\end{figure}

\item We sketch the graph of $y$ as follows:
\begin{figure}[H]\centering
\begin{tikzpicture}
\begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, domain=-2:2, samples=200]
\addplot[mark=none, color=blue, ultra thick,dashed] {exp(x)};
\addlegendentry{$y=e^x$};
\addplot[mark=none,color=green!60!black,ultra thick,dashed] {exp(-x)};
\addlegendentry{$y=e^{-x}$};
\addplot[mark=none,color=black,ultra thick,dashed] {0.5*exp(-x)};
\addlegendentry{$y=\frac{1}{2}\cdot e^{-x}$};
\addplot[mark=none,color=red,ultra thick] {1-0.5*exp(-x)};
\addlegendentry{$y=1-\frac{1}{2}\cdot e^{-x}$};
\end{axis}
\end{tikzpicture}
\end{figure}
\item The graph of $y=e^{|x|}$ is obtained by reflecting the portion of the graph of $y=e^x$ that lies to the left of $y$-axis onto the right of the $y$-axis while keeping the portion of the graph that is in the positive $x$ region unchanged:
\begin{figure}[H]\centering
\begin{tikzpicture}
\begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, domain=-2:2, samples=200]
\addplot[mark=none, color=blue, ultra thick,dashed] {exp(x)};
\addlegendentry{$y=e^x$};
\addplot[mark=none,color=red,ultra thick] {exp(abs(x))};
\addlegendentry{$y=e^{|x|}$};
\end{axis}
\end{tikzpicture}
\end{figure}
\end{enumerate}

Exercise 1.4.9 (9-14)

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Exercise 1.4.9 (9-14)

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