options: Use functions to model relationships between quantities.

Model periodic phenomena with trigonometric functions.

Build a function that models a relationship between two quantities.

Define, evaluate, and compare functions.

Extend the domain of trigonometric functions using the unit circle.

Understand the concept of a function and use function notation.

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

Construct and compare linear and exponential models and solve problems.

Prove and apply trigonometric identities.

Build new functions from existing functions.

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

Analyze functions using different representations.
problem_activity: Task
Carbon 14 is a form of carbon which decays exponentially over
time.
The amount
of Carbon 14 contained in a preserved plant is modeled
by the equation
$$
f(t) = 10\left(\frac{1}{2}\right)^{ct}.
$$
Time in this equation is measured in years
from the moment when the plant dies ($t = 0$) and the amount of Carbon 14 remaining in the preserved plant is measured in micrograms
(a microgram is one millionth of a gram).  The number $c$ in the exponential
measures the exponential rate of decay of Carbon $14$.
How many micrograms  of Carbon $14$ are in the plant at the time it died?
The best known estimate for the half-life of Carbon 14, that is the amount of time it takes for half of the
Carbon $14$ to decay, is $5730 \pm 40$ years.   Use this information to
calculate the range of possible values for the constant $c$ in the equation for $f$.
Use your answer from part (b) to find the range of years when there is
one microgram remaining in the preserved plant.


correct_option_index: 7