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.

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.
problem_activity: Warm-up
The goal of this warm-up is to review the meaning of a function presented graphically. While students do not need to use function notation here, interpreting the graph in terms of the context will prepare them for their work with functions in the rest of the unit.
Launch
Tell students to close their books or devices. Display the graph for all to see. Ask students to observe the graph and be prepared to share one thing they notice and one thing they wonder. Select students to briefly share one thing they noticed or wondered to help ensure all students understand the information conveyed in the graph. Ask students to open their books or devices and answer the questions about the graph. Follow with a whole-class discussion.
Student Facing
Here is a graph of the accumulated rainfall in Las Vegas, Nevada, in the first 60 days of 2017.
[IMG]
Use the graph to support your answers to the following questions.
Is the accumulated amount of rainfall a function of time?
Is time a function of accumulated rainfall?
Student Response
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Anticipated Misconceptions
If students struggle to see from the graph how the accumulated rainfall is a function of time but time is not a function of accumulated rainfall, consider displaying the data in a table. Shown here is the data for the first 20 days of 2017. Help students see that for every value of
\(t\)
, the time in days, there is one value of
\(r\)
, the accumulated rainfall in inches, but this is not true the other way around.
{"table": [{"_attributes": {"border": "1", "class": ["c-table-compressed"]}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "166"}, "_value": "\\(t\\)"}], "_value": "(days)"}], "td": [{"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "4"}, {"_value": "5"}, {"_value": "6"}, {"_value": "7"}, {"_value": "8"}, {"_value": "9"}, {"_value": "10"}, {"_value": "11"}, {"_value": "12"}, {"_value": "13"}, {"_value": "14"}, {"_value": "15"}, {"_value": "16"}, {"_value": "17"}, {"_value": "18"}, {"_value": "19"}, {"_value": "20"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "832"}, "_value": "\\(r\\)"}], "_value": "(inches)"}], "td": [{"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0.03"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.1"}, {"_value": "0.11"}, {"_value": "0.38"}]}]}]}]}
Activity Synthesis
Make sure students understand why the accumulated rain is a function of time but not the other way around.
Then, recall the notation for writing, using function notation, accumulated rain fall as a function of time. If
\(r\)
represents the amount of rainfall in inches and
\(t\)
is time in days, then
\(r(t)\)
is the amount of rain that has fallen in the first
\(t\)
days of 2017. For example,
\(r(2) = 0\)
tells us that the accumulated rainfall after in the first two days of the year was 0 inches.
\(r(48)=1\)
means that there was 1 inch of accumulated rain in the first 48 days of the year. Ask students to write and explain the meaning of a few other statements using function notation.


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