identify the graph of y = e^x - 2.

Explanation:

Step1: Analyze the parent - function

The parent - function of $y = e^{x}-2$ is $y = e^{x}$, which is an exponential growth function with a $y$-intercept at $(0,1)$ and passes through the points $(- 1,\frac{1}{e})\approx(-1,0.37)$ and $(1,e)\approx(1,2.72)$. The general form of an exponential function is $y = a\cdot b^{x}+k$, where for $y = e^{x}$, $a = 1$, $b = e\approx2.718$, and $k = 0$.

Step2: Consider the transformation

The function $y = e^{x}-2$ is a vertical shift of the function $y = e^{x}$ down by 2 units. The $y$-intercept of $y = e^{x}-2$ is obtained by substituting $x = 0$: $y=e^{0}-2=1 - 2=-1$. As $x\to-\infty$, $y = e^{x}-2\to - 2$ (since $\lim_{x\to-\infty}e^{x}=0$), and as $x\to+\infty$, $y = e^{x}-2\to+\infty$.

Step3: Match the graph

The graph of $y = e^{x}-2$ is an increasing exponential curve that crosses the $y$-axis at $(0, - 1)$ and has a horizontal asymptote at $y=-2$.

Answer:

The graph that has a $y$-intercept at $(0, - 1)$, is an increasing curve, and has a horizontal asymptote at $y = - 2$ (the second graph among the given options if we assume the order from left - to - right).