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, [tex]$40 \%$[/tex] of the sample remains alive.

Which best describes the graph of the function that represents the number of live bacteria after [tex]$x$[/tex] 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 :

To solve this problem, let’s break it down step-by-step and identify the function that models the bacterial population over time as well as its asymptotic behavior.

### Step 1: Initial Analysis
- We are given that the initial population of bacteria is 5,000.
- Each day, 40% of the bacteria remain alive, which implies that 60% of the bacteria die off each day.

### Step 2: Mathematical Representation
- The survival rate per day is 40%, which can be written as a decimal [tex]\(0.4\)[/tex].
- The number of bacteria remaining after [tex]\(x\)[/tex] days can be modeled using an exponential decay function.

### Step 3: Constructing the Function
- The initial amount of bacteria is [tex]\(5000\)[/tex].
- After [tex]\(1\)[/tex] day, the number of bacteria is [tex]\(5000 \times 0.4\)[/tex].
- After [tex]\(2\)[/tex] days, the number of bacteria is [tex]\(5000 \times (0.4)^2\)[/tex].
- After [tex]\(x\)[/tex] days, the number of bacteria is [tex]\(5000 \times (0.4)^x\)[/tex].

Thus, the function representing the number of live bacteria after [tex]\(x\)[/tex] days is:
[tex]\[ f(x) = 5000(0.4)^x \][/tex]

### Step 4: Analyzing the Asymptotes
- As [tex]\(x\)[/tex] increases, [tex]\( (0.4)^x \)[/tex] gets closer and closer to [tex]\(0\)[/tex] but never actually reaches it. This describes a horizontal asymptote.
- Therefore, the function has a horizontal asymptote at [tex]\( y = 0 \)[/tex].

### Step 5: Selection of the Correct Option
Now, let’s compare this with the given options:

1. [tex]\( f(x) = 5000(0.4)^x \)[/tex], with a horizontal asymptote of [tex]\( y = 0 \)[/tex]
2. [tex]\( f(x) = 5000(0.6)^x \)[/tex], with a vertical asymptote of [tex]\( x = 0 \)[/tex]
3. [tex]\( f(x) = 5000(1.4)^x \)[/tex], with a horizontal asymptote of [tex]\( y = 0 \)[/tex]
4. [tex]\( f(x) = 5000(1.6)^x \)[/tex], with a vertical asymptote of [tex]\( x = 0 \)[/tex]

Since we have determined that the function is [tex]\( f(x) = 5000(0.4)^x \)[/tex] with a horizontal asymptote at [tex]\( y = 0 \)[/tex], the correct option is:
[tex]\[ f(x) = 5000(0.4)^x, \text{ with a horizontal asymptote of } y = 0 \][/tex]

Thus, the best description for the graph of the function that represents the number of live bacteria after [tex]\(x\)[/tex] days of treatment is:
[tex]\[ f(x) = 5000(0.4)^x, \text{ with a horizontal asymptote of } y = 0 \][/tex]