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?

Question

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 \( b = 0 \).