options: Build new functions from existing functions.

Build a function that models a relationship between two quantities.

Analyze functions using different representations.

Use functions to model relationships between quantities.

Define, evaluate, and compare functions.

Understand the concept of a function and use function notation.

Model periodic phenomena with trigonometric functions.

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

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

Extend the domain of trigonometric functions using the unit circle.

Construct and compare linear and exponential models and solve problems.

Prove and apply trigonometric identities.
problem_activity: Activity
First, students examine several examples of functions and non-functions from the perspective of “Can we figure out . . . ?” Then, they decide which of the functions can be represented by a given expression.
Launch
Allow quiet think time to consider the situations in the first question and answer yes or no. Ensure students understand which ones are functions, and which are not. Then, give students time to complete the table.
Student Facing
For each question, answer yes or no.
It is 50 miles to Tucson. Can we figure out how many kilometers it is to Tucson?
It is 200 kilometers to Saskatoon. Can we figure out how many miles it is to Saskatoon?
A number is -3. Can we figure out its absolute value?
The absolute value of a number is 8. Can we figure out the number?
A circle has a diameter of 8 cm. Can we figure out its circumference?
A circle has a circumference of
\(10\pi\)
cm. Can we figure out its diameter?
A square has a side length of 6 units. Can we figure out its perimeter?
A rectangle has a perimeter of 30 meters. Can we figure out its width?
Which of the relationships are functions?
For each function definition in the table, match it with the situation, write a statement explaining which variable depends on which, and write an example using function notation. An example is done for you.
|    function definition     |                  situation                  |                                                      statement                                                      |    example    |
|----------------------------|---------------------------------------------|---------------------------------------------------------------------------------------------------------------------|---------------|
|     \(m(x) = 0.62x \)      | You know kilometers and want to find miles. | Distance in miles depends on distance in kilometers, or, distance in miles is a function of distance in kilometers. | \(m(100)=62\) |
| \(f(x) = x \boldcdot \pi\) |                                             |                                                                                                                     |               |
|      \(g(x) = 1.6x\)       |                                             |                                                                                                                     |               |
|       \(h(x) = 4x\)        |                                             |                                                                                                                     |               |
|       \(k(x) = |x|\)       |                                             |                                                                                                                     |               |
Student Response
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Activity Synthesis
Invite students to share their responses in the table, and how they know. Emphasize that there are many correct examples one could choose—the thing to understand is how function notation communicates an input and output to the function (or a value of the independent variable and corresponding value of the dependent variable.)


correct_option_index: 4