Question

Darnell is conducting a lab experiment where he introduces an antibiotic to bacteria. To begin the experiment, he had 1,000,000 CFU per milliliter of bacteria and found that the number of bacteria decreased at a rate of 22 percent per day. The situation can be modeled by the equation b(t)=1,000,000(0.78)^t, where b is the amount of bacteria after t days. Which one of the following statements about the key features of the exponential graph for this situation is true?
Statement #1: The graph of the function has a horizontal asymptote at t=0
Statement #2: The graph of the function has a horizontal asymptote at b=0
Statement #3: The graph of the function has a vertical asymptote at t=0
Statement #4: The graph of the function has a vertical asymptote at b=0

Answer 1

Let's analyze the given statements regarding the key features of the exponential graph that models the decrease in bacteria over time.

The function is given by:

\[ b(t) = 1,000,000 (0.78)^t \]

### Statement #1: The graph of the function has a horizontal asymptote at \( t = 0 \).
This statement is **false**. The horizontal asymptote refers to the behavior of the function as \( t \) approaches infinity, not at \( t = 0 \).

### Statement #2: The graph of the function has a horizontal asymptote at \( b = 0 \).
This statement is **true**. As \( t \) approaches infinity, the value of \( b(t) \) approaches 0. Hence, the graph has a horizontal asymptote at \( b = 0 \).

### Statement #3: The graph of the function has a vertical asymptote at \( t = 0 \).
This statement is **false**. A vertical asymptote describes behavior as \( t \) approaches a specific value from the left or right, and in this case, \( t = 0 \) is not where the function approaches infinity or undefined.

### Statement #4: The graph of the function has a vertical asymptote at \( b = 0 \).
This statement is **false**. Vertical asymptotes occur in cases where the function value approaches infinity, but here, as \( b(t) \) can never reach or become negative, it does not have a vertical asymptote at any point.

### Summary:
Only **Statement #2** is true. The graph of the function has a horizontal asymptote at \( b = 0 \).