options: 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.

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.
problem_activity: Activity
If students also use technology to create the tables, they are using appropriate tools strategically (MP5).
Launch
Provide access to graphing technology. Students may need some assistance adjusting their graphing window to see the relevant features of each graph. Either help them set the window before they start to a domain of
\(\text-2 \le x \le 8\)
and range of
\(\text-60 \le y \le 600\)
, or wait until they have attempted to graph at least one function and remind them of what they have learned about changing the graphing window.
Student Facing
Each function represents the amount in a bank account after
\(t\)
weeks.
\(A(t) = 500\)
\(B(t) = 500 + 40t\)
\(C(t) = 500 - 40t\)
\(D(t) = 500 \boldcdot (1.5)^t\)
\(E(t) = 500 \boldcdot (0.75)^t\)
Make a table for each bank account showing the money in the account at 0, 1, 2, and 3 weeks.
Describe in words how the money in the account is changing week by week.
Use technology to create a graph of each function. How can you see your description in each graph?
Student Response
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Activity Synthesis
Focus the discussion on the connection between the equations, descriptions and graphs. Discuss questions such as:
“What do all functions have in common?” (The vertical intercept is 500, each bank account starts with $500)
“How does the money change each week? How do we see this change in the graph?"
“How are
\(B\)
and
\(C\)
the same and different?” (They are the same because they are each linear, and each start with
$
500 and change by
$
40 per week. They are different because B increases and C decreases (or the rate of change is positive versus negative.)
“How are
\(D\)
and
\(E\)
the same and different?” (They are the same because they are each exponential, and each start with
$
500. They are different because
\(D\)
increases and
\(E\)
decreases (or the growth factor is greater than 1 versus less than 1.))


correct_option_index: 1