practice graphing exponential functions in base e. identify the graph of y = e^x - 2.

Explanation:

Step1: Analyze the parent - function

The parent function of $y = e^{x}-2$ is $y = e^{x}$. The function $y = e^{x}$ has a $y$ - intercept at $(0,1)$ (since when $x = 0$, $y=e^{0}=1$) and passes through the points $(- 1,\frac{1}{e})\approx(-1,0.37)$ and $(1,e)\approx(1,2.72)$. It has a horizontal asymptote at $y = 0$.

Step2: Analyze the transformation

The function $y = e^{x}-2$ is a vertical translation of the function $y = e^{x}$ down by 2 units. The $y$ - intercept of $y = e^{x}-2$ is obtained by substituting $x = 0$: $y=e^{0}-2=1 - 2=-1$. The horizontal asymptote of $y = e^{x}-2$ is $y=-2$.

Step3: Match the graph

The graph of $y = e^{x}-2$ will be an increasing exponential curve with a $y$ - intercept at $(0, - 1)$ and a horizontal asymptote at $y=-2$.

Answer:

The graph that has a $y$ - intercept at $(0,-1)$ and is an increasing exponential function with a horizontal asymptote at $y = - 2$ (the first graph among the options if we assume the order from left - to - right as described in the problem).