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

Explanation:

Step1: Recall exponential decay formula

The general form of an exponential decay function is \( f(x) = a(b)^x \), where \( a \) is the initial amount, \( b \) is the decay factor (\( 0 < b < 1 \)), and \( x \) is the time. Here, the initial population \( a = 5000 \). After each day, 40% remains alive, so the decay factor \( b = 0.4 \) (since 40% = 0.4). So the function should be \( f(x)=5000(0.4)^x \).

Step2: Analyze asymptotes of exponential functions

Exponential functions of the form \( f(x) = a(b)^x \) (where \( a
eq 0 \), \( b > 0 \), \( b
eq 1 \)) have a horizontal asymptote at \( y = 0 \). Vertical asymptotes occur for functions like rational functions (e.g., \( \frac{1}{x} \) has a vertical asymptote at \( x = 0 \)), but exponential functions do not have vertical asymptotes. So we check the asymptote: the function \( f(x)=5000(0.4)^x \) has a horizontal asymptote at \( y = 0 \).

Answer:

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