# Exponential Parameters

Alignments to Content Standards: F-BF.A F-LE.A.2 A-SSE.A.1 A-SSE.A.1.a

1. In a carefully controlled biology lab, a population of 100 bacteria reproduces via binary fission. That is, every hour, on the hour, each bacteria splits into two bacteria. Assuming no bacteria deaths, find an expression for the number $P(t)$ of bacteria in the population after $t$ hours.
2. In the next lab over, a population of protists reproduces hourly according to multiple fission. The function which gives the population of protists after $t$ hours is $$P(t)=50\cdot 3^t.$$ Interpret the significance of the numbers 50 and 3 in the context of the biological experiment.

## IM Commentary

The task provides a reasonably straight-forward introduction to interpreting the parameters of an exponential function in terms of a modeling context. In general, an exponential function $f(t)=ab^t$ has two parameters. The parameter $a$ is interpreted as the starting value (when $t$ represents time), and $b$ represents the growth rate -- the amount the quantity is multiplied by each time the value of $t$ is incremented by 1. The task has students both generate an exponential expression from a contextual description, and in reverse, interpret parameters in a context from an algebraic expression.

The task involves some words from biology that might be intimidating to students.  Teachers should be prepared to explain that bacteria and protists are very simple single-celled creatures, and clarfiy and confusion around fission -- binary (or multiple) fission is a means of reproduction in which an organism splits into two (or multiple) copies.  Alternatively, the task might make a nice connection to students' existing work in a science class.

## Solution

1. The population starts with 100 members. After 1 hour, there are 200 bacteria. After 2 hours, there are 400 bacteria. After $t$ hours, the initial population has doubled $t$ times, giving us $P(t)=100\cdot 2^t$ members.
2. We observe that $P(0)=50\cdot 3^0=50$, so a concise interpretation of the number 50 is the initial population of the protists (the number of protists at the start of the experiment). Now by evaluating $P(1)=150$, $P(2)=450$, etc., we see that the population triples every hour. In other words, the expression $50\cdot 3^t$ can be thought of as 50 multiplied by 3 exactly $t$ times, once per hour. So a concise interpretation of the constant 3 is the factor by which the population grows every hour.