A population of bacteria is treated with an antibiotic. It is estimated that 5,000 live bacteria existed in the sample before treatment. After each day of treatment, $40 \%$ of the sample remains alive. Which best describes the graph of the function that represents the number of live bacteria after $x$ days of treatment?
A. [tex]f(x)=5000(0.4)^x[/tex], with a horizontal asymptote of [tex]y=0[/tex]
B. [tex]f(x)=5000(0.6)^x[/tex], with a vertical asymptote of [tex]x=0[/tex]
C. [tex]f(x)=5000(1.4)^x[/tex], with a horizontal asymptote of [tex]y=0[/tex]
D. [tex]f(x)=5000(1.6)^x[/tex], with a vertical asymptote of [tex]x=0[/tex]
Answer (2)
The function representing the number of live bacteria after x days is f ( x ) = 5000 ( 0.4 ) x .
As x approaches infinity, f ( x ) approaches 0, indicating a horizontal asymptote at y = 0 .
There is no vertical asymptote.
The best description is f ( x ) = 5000 ( 0.4 ) x , with a horizontal asymptote of y = 0 .
Explanation
Understanding the Problem We are given that the initial population of bacteria is 5,000. After each day, 40% of the bacteria remain alive. We need to find the function that represents the number of live bacteria after x days of treatment and identify its key features (asymptotes).
Defining the Function Let f ( x ) be the number of live bacteria after x days. Since 40% of the bacteria remain alive each day, the decay factor is 0.4. Therefore, the function can be expressed as f ( x ) = 5000 ( 0.4 ) x .
Analyzing Asymptotes Now, let's analyze the function f ( x ) = 5000 ( 0.4 ) x to determine its asymptotes. As x approaches infinity, ( 0.4 ) x approaches 0, so f ( x ) approaches 0. This indicates a horizontal asymptote at y = 0 . The function is defined for all non-negative x values. There is no vertical asymptote.
Selecting the Best Description Comparing the derived function and asymptote with the given options, we can see that the best description is f ( x ) = 5000 ( 0.4 ) x , with a horizontal asymptote of y = 0 .
Examples
Understanding exponential decay is crucial in various real-world scenarios. For instance, when administering medication, the concentration of the drug in the bloodstream decreases over time. Similarly, in finance, the value of an asset can depreciate exponentially. By modeling these phenomena with exponential functions, we can make informed decisions about dosage, investment strategies, and resource management.
The function representing the number of live bacteria after x days of treatment is f ( x ) = 5000 ( 0.4 ) x , with a horizontal asymptote at y = 0 . The best answer choice is A . This captures the exponential decay trend as the bacteria diminish due to the antibiotic treatment.
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