Question

which of the following is the graph of $y = e^x + 1$? three graphs are shown with radio buttons

Explanation:

Step1: Recall properties of \( y = e^x + 1 \)

The function \( y = e^x \) has a horizontal asymptote at \( y = 0 \) (as \( x \to -\infty \), \( e^x \to 0 \)) and passes through \( (0, 1) \). For \( y = e^x + 1 \), the horizontal asymptote shifts up by 1 unit, so the asymptote is \( y = 1 \). Also, when \( x = 0 \), \( y = e^0 + 1 = 2 \), so the graph passes through \( (0, 2) \).

Step2: Analyze the graphs

• First graph: Check the y - intercept. If it's near \( y = 1 \) or \( y = 2 \)? The first graph seems to have a y - intercept around \( y = 1 \) (or lower), not matching \( (0, 2) \).
• Second graph: Check the horizontal asymptote (should be \( y = 1 \)) and the y - intercept (should be \( (0, 2) \)). The second graph has a horizontal asymptote around \( y = 1 \) (or the behavior as \( x\to-\infty \) approaches \( y = 1 \)) and passes through \( (0, 2) \) (visually, the y - intercept is at \( y = 2 \)).
• Third graph: The third graph has a different shape (increasing for negative x? No, \( y = e^x+1 \) is increasing for all x, but the third graph seems to be increasing from the left - hand side in a way that doesn't match the exponential function's behavior. Also, the horizontal asymptote and y - intercept don't match.

Answer:

The Middle Graph (the second graph among the three)