options: Understand the concept of a function and use function notation.

Construct and compare linear and exponential models and solve problems.

Build a function that models a relationship between two quantities.

Interpret expressions for functions in terms of the situation they model.

Prove and apply trigonometric identities.

Use functions to model relationships between quantities.

Interpret functions that arise in applications in terms of the context.

Model periodic phenomena with trigonometric functions.

Build new functions from existing functions.

Define, evaluate, and compare functions.

Analyze functions using different representations.

Extend the domain of trigonometric functions using the unit circle.
problem_activity: Task
An important example of a model often used in biology or ecology to model
population growth is called the logistic growth model.  The general
form of the logistic equation is
$$
P(t) = \frac{KP_0e^{rt}}{K+P_0(e^{rt}-1)}.
$$
In this equation $t$ represents time, with $t = 0$ corresponding to when the population in question is first measured; $K,P_0$  and $r$ are all real numbers
with $K$ being called the ''carrying capacity'' while $r$ is a growth rate and
is normally a positive number.
Explain why the value $P_0$ represents
the  population when it is first measured.
Explain why, as time elapses, the population stabilizes, approaching the value $K$.
Explain how the behavior of $P$ changes if the growth rate $r$ is increased or decreased.
Below is the graph of a particular logistic function $P$, showing the growth of a
bacteria population.  Using the graph, identify $P_0$ and  $K$.
[IMG]
Using the values of $P_0$ and $K$ from the previous part, sketch the graph of the logistic function $Q$ given by
$$
Q(t) =  \frac{KP_0e^{2rt}}{K+P_0(e^{2rt}-1)}.
$$
Note that $Q$ is the same as $P$ except that the growth rate $r$ has been doubled.


correct_option_index: 6