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 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

In the given situation with the equation \( b(t) = 1,000,000 (0.78)^t \), let's analyze the statements regarding the key features of the exponential graph.

1. Statement #1: The graph of the function has a horizontal asymptote at \( t=0 \).

   • This statement is false. Horizontal asymptotes are related to the behavior of the function as \( t \) approaches infinity, not a specific value of \( t \).

2. 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. Thus, the horizontal asymptote is at \( b = 0 \).

3. Statement #3: The graph of the function has a vertical asymptote at \( t=0 \).

   • This statement is false. Vertical asymptotes typically occur where the function is undefined. In this case, the function is defined for \( t = 0 \) and does not have a vertical asymptote.

4. Statement #4: The graph of the function has a vertical asymptote at \( b=0 \).

   • This statement is false. Similar to the previous reasoning, a vertical asymptote cannot occur at a certain value of \( b\). The function does not become undefined at \( b = 0 \).

Therefore, among the statements given, only Statement #2 is true.