options: Build new functions from existing functions.

Define, evaluate, and compare functions.

Use functions to model relationships between quantities.

Understand the concept of a function and use function notation.

Construct and compare linear and exponential models and solve problems.

Analyze functions using different representations.

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

Build a function that models a relationship between two quantities.

Prove and apply trigonometric identities.

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

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.
problem_activity: Activity
The purpose of this activity is to show how a context can impose restrictions on the domain and range of a function. Given a description of a context, students determine the restrictions on the domain and range imposed by the context. Considering how a context restricts the domain or range of a function modeling it is an important part of mathematical modeling (MP4). This prepares students to interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson.
Launch
Remind students that domain refers to possible values of the independent variable, and range refers to possible values of the dependent variable. Use a few examples from the previous activity to illustrate. For example, for the cab ride, the domain is all numbers that are greater than 0, and the range is all numbers that are greater than 3.5. This presumes that a ride must have some distance (it can’t be 0 miles long), and can be any number of miles long. A modeler might decide not to consider any cab rides that are longer than, say, 100 miles, in which case the domain would be all numbers between 0 and 100. Sometimes restrictions on the domain are a decision made by the modeler. In contrast, for the function modeling the admission price to the park based on number of people, the domain only contains whole, positive numbers, since the number of people must be a whole, positive number.
Arrange students in groups of 2. After a few minutes of quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they look different. Follow with a whole-class discussion.
Student Facing
To make sense in a given context, many functions need restrictions on the domain and range. For each description of a function
describe the domain and range
describe what its graph would look like (separate dots, or connected?)
weight of a puppy as a function of time
number of winter coats sold in a store as a function of temperature outside
number of books in a library as a function of number of people who live in the community the library serves
height of water in a tank as a function of volume of water in the tank
amount of oxygen in the atmosphere as a function of elevation above or below sea level
thickness of a folded piece of paper as a function of number of folds
Student Response
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Activity Synthesis
Much discussion takes place between partners. Invite students to share how they determined the domain and range of the function representing the context. Ask if other groups have different responses. Choose as many different groups as time allows. Attend to the language that students use to describe their pairing, giving them opportunities to describe their relationships more precisely.


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