Question

practice graphing exponential functions in base e.

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

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Identify the horizontal asymptote

The parent function is \( y = e^x \), which has a horizontal asymptote at \( y = 0 \).

Subtracting \( 2 \) shifts the entire graph down by \( 2 \) units:

$$ y = e^x - 2 $$

This shifts the horizontal asymptote to:

$$ y = -2 $$

Looking at the options:

• The first graph has a horizontal asymptote at \( y = 0 \).
• The second graph has a horizontal asymptote at \( y = -2 \).
• The third graph has a horizontal asymptote at \( y = -2 \).

Step 2: Find the y-intercept

To find where the graph crosses the y-axis, substitute \( x = 0 \) into the equation:

$$ y = e^0 - 2 $$

$$ y = 1 - 2 $$

$$ y = -1 $$

The y-intercept is at \( (0, -1) \).

Looking at the remaining options with asymptote \( y = -2 \):

• The second graph passes through \( (0, -1) \) and curves upwards from left to right (exponential growth).
• The third graph shows exponential decay.

Therefore, the second graph is the correct representation.

Answer:

The correct graph is the second option (the middle graph), which has a horizontal asymptote at \( y = -2 \) and a y-intercept at \( (0, -1) \).