Question

a science experiment begins with a bacterial population of 12. after 1 hour, the population is 18. after 2 hours, the population is 27. which best describes the relationship between the time, in hours, and the population of the bacteria? what is the y - intercept of the function? what is the rate of change of the function?

Explanation:

Step1: Determine the relationship type

We check if it's linear or exponential. For linear, the difference in population per - unit time should be constant. For exponential, the ratio of populations at consecutive time - points should be constant.
Initial population $P_0 = 12$, population at $t = 1$ is $P_1=18$, population at $t = 2$ is $P_2 = 27$.
The ratio $\frac{P_1}{P_0}=\frac{18}{12}=\frac{3}{2}$, and $\frac{P_2}{P_1}=\frac{27}{18}=\frac{3}{2}$. So it's an exponential relationship.

Step2: Find the y - intercept

The y - intercept of a function representing the population over time is the initial population. When $t = 0$, the population is 12. So the y - intercept is 12.

Step3: Find the rate of change

For an exponential function of the form $y = a\cdot b^x$, where $a$ is the initial value and $b$ is the growth factor. Here $a = 12$ and $b=\frac{3}{2}$. The rate of change of an exponential function $y = a\cdot b^x$ is given by the growth factor $b$. So the rate of change is $\frac{3}{2}$.

Answer:

1. Exponential relationship
2. 12
3. $\frac{3}{2}$