Lesson 8: Rising and Falling
About this lesson
This lesson marks the beginning of a transition for students and their thinking about periodic functions. Previously, students approached the idea of periodic functions in the context of circular motion, such as the motion of the end of a minute hand on a clock. In this lesson, we expand the idea of a periodic function to include any function in which the output values repeat at regular intervals and show that the graphs of these types of functions can have a wavelike appearance.
During this lesson, students have the opportunity to communicate their observations about graphs precisely with others (MP6). In addition, they observe repeated behavior in graphs by contrasting them with graphs that do not repeat, which has been the norm for graphs up until this unit.
Lesson overview
 8.1 Warmup: Notice and Wonder: A Bouncing Curve (5 minutes)

8.2 Activity: What is Happening? (15 minutes)
 Includes "Are you Ready for More?" extension problem
 8.3 Activity: Card Sort: Graphs of Functions (15 minutes)
 Lesson Synthesis
 8.4 Cooldown: Measuring the Hours (5 minutes)
Learning goals:
 Compare and contrast (in writing) the features of the cosine, sine, and tangent functions.
Learning goals (student facing):
 Let’s study graphs that repeat.
Learning targets (student facing):
 I understand that the graph of a periodic function can look like a wave whose outputs repeat between the same maximum and minimum values.
Required materials:
 Preprinted slips, cut from copies of the blackline master
Glossary:
 periodic function  A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).
 Access the complete Algebra 2 glossary.
Standards:
 This lesson builds towards the standard: CCSS.HSFTF.BMS.FTF
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.