To solve this problem, let's break it down step-by-step.
1. Initial Population of Bacteria:
- The initial number of bacteria before treatment is given as 5,000.
2. Daily Decay Rate:
- After each day of treatment, 40% of the bacteria remain alive. This implies that [tex]\(0.4\)[/tex] (or 40%) of the bacteria survive each day.
3. Decay Function:
- To find the number of bacteria remaining after [tex]\(x\)[/tex] days of treatment, we need to set up an exponential decay function. The general form of the exponential decay function is:
[tex]\[
f(x) = A \cdot (r)^x
\][/tex]
where:
- [tex]\(A\)[/tex] is the initial quantity (in this case, 5,000),
- [tex]\(r\)[/tex] is the decay rate (in this case, 0.4),
- [tex]\(x\)[/tex] is the time in days.
4. Forming the Function:
- Substituting the given values into the formula, we get the function:
[tex]\[
f(x) = 5000 \cdot (0.4)^x
\][/tex]
5. Analyzing the Asymptote:
- As [tex]\(x\)[/tex] (the number of days) increases, [tex]\((0.4)^x\)[/tex] gets smaller and smaller, approaching 0. Thus, the function [tex]\(f(x) = 5000 \cdot (0.4)^x\)[/tex] gets closer and closer to 0, but never actually reaches 0.
- This means that the graph of the function has a horizontal asymptote at [tex]\(y = 0\)[/tex].
6. Conclusion:
- Given the decay rate and the asymptote, the best description of the graph of the function is:
[tex]\[
f(x) = 5000 \cdot (0.4)^x, \text{ with a horizontal asymptote of } y = 0
\][/tex]
Hence, the correct description is:
[tex]\[
f(x) = 5000(0.4)^x, \text{ with a horizontal asymptote of } y = 0
\][/tex]
This matches the first option provided in the question.
