options: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.  For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
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: 1