Question

an antibiotic is added to a bacteria culture. the function, n(t), represents the count of bacteria alive with respect to time in seconds in an exponential manner.
n(t) = 8,000(0.79)^t
graph of the function n(t) with t (time in seconds) on the x - axis and population of bacteria on the y - axis
what is the asymptote and y - intercept of the function that represents the count of bacteria alive?

Explanation:

Step1: Identify asymptote of exponential decay

For \( n(t) = ab^t \) (\( 0 < b < 1 \)), as \( t \to \infty \), \( n(t) \to 0 \). So asymptote is \( n(t) = 0 \).

Step2: Calculate y-intercept

Set \( t = 0 \): \( n(0) = 8000(0.79)^0 = 8000 \times 1 = 8000 \). Y-intercept is \( (0, 8000) \).

Answer:

Asymptote: \( t \)-axis (\( n(t) = 0 \)); y-intercept: \( (0, 8000) \)