Question

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

Explanation:

Step1: Recall properties of \( y = e^x \)

The parent function \( y = e^x \) has a y - intercept at \( (0,1) \), is increasing, and has a horizontal asymptote at \( y = 0 \) (as \( x
ightarrow-\infty \)).

Step2: Analyze transformation for \( y = e^x-2 \)

The function \( y = e^x - 2 \) is a vertical shift of \( y = e^x \) down by 2 units. So the y - intercept of \( y = e^x-2 \) is at \( (0,1 - 2)=(0,-1) \), the horizontal asymptote is \( y=-2 \) (as \( x
ightarrow-\infty \)), and the function is still increasing.

Step3: Analyze the graphs

• First graph: Y - intercept at \( (0,0) \), horizontal asymptote \( y = 0 \). Doesn't match \( y = e^x-2 \).
• Second graph: Y - intercept around \( (0,-1) \), horizontal asymptote \( y=-2 \) (as \( x

ightarrow-\infty \)), and increasing. Matches the properties of \( y = e^x-2 \).

• Third graph: The function is decreasing (since it's a reflection or a different exponential form like \( y = e^{-x} \) type), so it doesn't match \( y = e^x-2 \) which is increasing.

Answer:

The Middle Graph