starting with the graph of y = e^x, use transformations to sketch the graph of the function and state its…

Explanation:

Step1: Analyze the transformation of $y = e^x$ to $y = e^{-x}$

The transformation from $y = e^x$ to $y=e^{-x}$ is a reflection about the y - axis. For any point $(x,y)$ on $y = e^x$, the corresponding point on $y = e^{-x}$ is $(-x,y)$.

Step2: Analyze the transformation of $y = e^{-x}$ to $y = e^{-x}-1$

The transformation from $y = e^{-x}$ to $y = e^{-x}-1$ is a vertical shift down by 1 unit. For any point $(x,y)$ on $y = e^{-x}$, the corresponding point on $y = e^{-x}-1$ is $(x,y - 1)$.

Step3: Find the horizontal asymptote

The general form of an exponential function is $y = a\cdot b^{x - h}+k$. For the function $y = e^{-x}-1$, as $x\to+\infty$, $e^{-x}=\frac{1}{e^{x}}\to0$. So $y = e^{-x}-1\to - 1$. The horizontal asymptote is $y=-1$.

Answer:

The graph of $y = e^x$ is reflected about the y - axis and then shifted down 1 unit. The horizontal asymptote is $y = - 1$.