options: Use functions to model relationships between quantities.

Build a function that models a relationship between two quantities.

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

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

Construct and compare linear and exponential models and solve problems.

Understand the concept of a function and use function notation.

Define, evaluate, and compare functions.

Build new functions from existing functions.

Prove and apply trigonometric identities.

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

Analyze functions using different representations.
problem_activity: Optional activity
In this activity, students investigate the average rates of change of the three loan options from an earlier activity. Calculating average rates of change over the same interval of time but at different points in the lending period (early and then later) further reinforces the dramatic change that can result from compounding over time.
Launch
Recall the previous work students have done on the three loans (A, B, C) and tell students that they are going to examine how these loans change over different time intervals.
Student Facing
The functions
\(a\)
,
\(b\)
, and
\(c\)
represent the amount owed (in dollars) for Loans A, B, and C respectively: the input for the functions is
\(t\)
, the number of years without payments.
For each loan, find the average rate of change per year between:
the start of the loan and the end of the second year
the end of the tenth year and the end of the twelfth year
How do the average rates of change for the three loans compare in each of the two-year intervals?
Student Response
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Activity Synthesis
Select students to share their calculations for the average rates of change for each loan. Invite students to share their observations about how the different average rates of change compare. If not mentioned by students, ask them to quantify how the average rate of change increased between the two intervals for the different loans, such as by calculating how many times larger the value for the second interval is when compared the first.


correct_option_index: 10