options: Extend the domain of trigonometric functions using the unit circle.

Analyze functions using different representations.

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

Build new functions from existing functions.

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

Use functions to model relationships between quantities.

Build a function that models a relationship between two quantities.

Prove and apply trigonometric identities.

Model periodic phenomena with trigonometric functions.

Understand the concept of a function and use function notation.

Construct and compare linear and exponential models and solve problems.

Define, evaluate, and compare functions.
problem_activity: Warm-up
The purpose of this warm-up is for students to use repeated reasoning to write an algebraic expression to represent a rule of a function (MP8). The whole-class discussion should focus on the algebraic expression in the final row, however the numbers in the table give students an opportunity to also practice calculating the square of numbers written in fraction and decimal form.
Launch
Arrange students in groups of 2. Give students 1–2 minutes of quiet work time and then time to share their algebraic expression with their partner. Follow with a whole-class discussion.
Student Facing
Fill in the table of input-output pairs for the given rule. Write an algebraic expression for the rule in the box in the diagram.
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "input"}, {"_attributes": {"scope": "col"}, "_value": "output"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "8"}, {}]}, {"td": [{"_value": "2.2"}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "40402"}, "_value": "\\(12\\frac14\\)"}]}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}, {}]}]}]}]}
Student Response
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Activity Synthesis
Select students to share how they found each of the outputs. After each response, ask the class if they agree or disagree. Record and display responses for all to see. If both responses are not mentioned by students for the last row, tell students that we can either put
\(s^2\)
or
\(A\)
there. Tell students we can write the equation
\(A = s^2\)
to represent the rule of this function.
End the discussion by telling students that while we’ve used the terms input and output so far to talk about specific values, when a letter is used to represent any possible input we call it the
independent variable
and the letter used to represent all the possible outputs is the
dependent variable.
Students may recall these terms from earlier grades. In this case,
\(s\)
is the independent variable and
\(A\)
the dependent variable, and we say “
\(A\)
depends on
\(s\)
.”


correct_option_index: 11