Question

practice graphing exponential functions in base e.

what is the graph of (y = e^x - 2)?

Explanation:

⚡ Using what you learned: Graphing Exponential Functions

Step 1: Identify the function and its asymptote

The problem asks to graph the exponential function:

$$ y = e^x - 1 $$

The parent function is \( y = e^x \), which has a horizontal asymptote at \( y = 0 \).
Subtracting \( 1 \) shifts the entire graph down by \( 1 \) unit. Therefore, the horizontal asymptote of \( y = e^x - 1 \) is:

$$ y = -1 $$

Step 2: Find key points on the graph

Let's evaluate the function at a few key values of \( x \):

• At \( x = 0 \):

$$ y = e^0 - 1 = 1 - 1 = 0 $$

So, the graph must pass through the origin \( (0, 0) \).

• At \( x = 1 \):

$$ y = e^1 - 1 \approx 2.718 - 1 = 1.718 $$

So, the graph passes through approximately \( (1, 1.72) \).

Step 3: Match with the correct option

• First graph: Has a horizontal asymptote at \( y = 0 \) and passes through \( (0, 1) \). This is the graph of \( y = e^x \).
• Second graph: Has a horizontal asymptote at \( y = -1 \) (indicated by the dashed line at \( y = -1 \)) and passes through the origin \( (0, 0) \). This matches our function \( y = e^x - 1 \).
• Third graph: Shows a decaying exponential function, which does not match our base \( e > 1 \).

Answer:

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