Exercise 1.4.5 (5-8)

Question

Graph the given functions on a common screen. How are these graphs related?

\begin{enumerate}
  \setcounter{enumi}{4}
  \item $y=2^{x}, \quad y=e^{x}, \quad y=5^{x}, \quad y=20^{x}$

\item $y=e^{x}, \quad y=e^{-x}, \quad y=8^{x}, \quad y=8^{-x}$

\item $y=3^{x}, \quad y=10^{x}, \quad y=\left(\frac{1}{3}\right)^{x}, \quad y=\left(\frac{1}{10}\right)^{x}$

\item $y=0.9^{x}, \quad y=0.6^{x}, \quad y=0.3^{x}, \quad y=0.1^{x}$

\end{enumerate}

Khagan Gulusoy · Accepted Answer

All of these functions are exponential functions and therefore, they all cross the point $(0,1)$.

\begin{enumerate}
  \setcounter{enumi}{4}
  \item \href{https://www.desmos.com/calculator/xwvzed6zv2?embed}{https://www.desmos.com/calculator/xwvzed6zv2?embed}

The larger $b$, the faster does the function increase as $x \rightarrow \infty$ and the faster it approaches $0$ as $x \rightarrow -\infty$.\end{enumerate}

\begin{enumerate}
  \setcounter{enumi}{5}
  \item \href{https://www.desmos.com/calculator/twwxfl9zyr?embed}{https://www.desmos.com/calculator/twwxfl9zyr?embed}

The graph of $e^{x}$ and $8^{x}$ is the reflection of the graph of $e^{-x}$ and $8^{x}$ respectively.\end{enumerate}

\begin{enumerate}
  \setcounter{enumi}{6}
  \item \href{https://www.desmos.com/calculator/efvwx1z3vj?embed}{https://www.desmos.com/calculator/efvwx1z3vj?embed}

The functions with a base greater than $1$ are increasing and the functions with a base less than $1$ are decreasing.\end{enumerate}

\begin{enumerate}
  \setcounter{enumi}{7}
  \item \href{https://www.desmos.com/calculator/sljravnenf?embed}{https://www.desmos.com/calculator/sljravnenf?embed}

Each graph approaches $\infty$ as $x \rightarrow-\infty$ and 0 as $x \rightarrow \infty$.\end{enumerate}

Exercise 1.4.5 (5-8)

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Exercise 1.4.5 (5-8)

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