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.

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.

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.
problem_activity: Problem 1
A ball is thrown straight up from the top of a
$${48}$$
-foot tall building with an initial speed of
$${32}$$
feet per second. The height of the ball as a function of time can be modeled by the function
$$h(t)=-16t^2+{32}t+{48}$$
. Below is a graph and table of the function.
[IMG]
a. What are the solutions to this function? Interpret each solution in context of the situation.
b. What is the vertex of this function? Interpret the vertex in context of the situation.
Problem 2
The table below represents the value of Andrew’s stock portfolio, where
$$V$$
represents the value of the portfolio in hundreds of dollars and
$$t$$
is the time in months since he started investing. Answer the questions that follow based on the table of values.
| $$t$$                    (months) | $$V(t)$$                    (hundreds of dollars) |
|-----------------------------------|---------------------------------------------------|
|                 2                 |                        325                        |
|                 4                 |                        385                        |
|                 6                 |                        405                        |
|                 8                 |                        385                        |
|                10                 |                        325                        |
|                12                 |                        225                        |
|                14                 |                        85                         |
|                16                 |                        -95                        |
|                18                 |                       -315                        |
Assuming this data is quadratic, how much did Andrew invest in his stock initially? Explain how you arrived at this answer.
What is the maximum value of his stock and how long did it take to reach the maximum value?
After approximately how many months is Andrew’s stock portfolio valued at $0?
How fast is Andrew’s stock value decreasing between months 10 and 12? Find a two-month interval where the average rate of change is faster than it is between months 10 and 12 and explain why.
Problem 3
John is building a small rectangular pen for chickens. He wants the length to be 6 more feet than twice the width.
If the area of the pen is 36 square feet, what are the dimensions of the pen in feet?


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