Question

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? f(x) = 5000(0.4)^x, with a horizontal asymptote of y = 0 f(x) = 5000(0.6)^x, with a vertical asymptote of x = 0 f(x) = 5000(1.4)^x, with a horizontal asymptote of y = 0 f(x) = 5000(1.6)^x, with a vertical asymptote of x = 0

Explanation:

Step1: Identify the exponential - decay formula

The general form of an exponential - decay function is $y = a(1 - r)^x$, where $a$ is the initial amount, $r$ is the rate of decay, and $x$ is the number of time - periods. Here, $a = 5000$ (the initial number of bacteria) and $r=0.6$ (since 40% remains alive, 60% is killed, so the rate of decay is 0.6). The function for the number of live bacteria after $x$ days is $f(x)=5000\times(0.4)^x$.

Step2: Analyze the asymptote of the exponential - decay function

For an exponential - decay function of the form $y = a(b)^x$ where $0 < b<1$ ($a>0$), as $x
ightarrow+\infty$, $y
ightarrow0$. So the horizontal asymptote of the function $y = 5000\times(0.4)^x$ is $y = 0$.

Answer:

A. $f(x)=5000(0.4)^x$, with a horizontal asymptote of $y = 0$