# Period and Amplitude

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## Objective

Students will be able to identify the period and amplitude of a sine or cosine graph or equation.

#### Big Idea

What makes the graph of a trig function unique? Find out in this lesson.

## Warm-up and Homework Review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Up- Period and Amplitude, which asks students to compare trigonometric functions to other functions studies this year.

I also use this time to correct and record the previous day's Homework.

## Reviewing the graphs of sine and cosine

10 minutes

In the previous lesson, students constructed both the sine and cosine graph from a unit circle.  The first goal in today's lesson is to draw these graphs in their notes along with all of the major features (except period and amplitude which are the topics of today's lesson).  Their graphs need to include the height of each function, the local maximum and minimum values, the y-intercept, and the x-intercepts.  For the re-occurring points, this is an important place to discuss how we can write down an expression that represents every one of those points (Math Practice 8) I have the students discuss in pairs how you can represent these reoccurring points.  The goal is that they come up with ±2π.

## Understanding Amplitude

5 minutes

The next section has students look at the amplitude as the height of one of these graphs and then gives the students some examples of finding amplitude.  Please note that I am using both degrees and radians as the unit for the x-axis.  This practice will reinforce their understanding of the connections between degrees and radians (Math Practice 7).

## Understanding Period

15 minutes

Period is a bit more involved than amplitude.  Instead of giving the students the formula for finding period and having them apply it, we are going to give them an opportunity to build the concept of period as a horizontal stretch or shrink.  The first four examples all include both the equation and the graph.  The graphs can come up after the student discuss or during as needed.  This is a major scaffolding piece in this lesson.  The next examples just include the equation.  By the end, have the students write a statement on how to find period.

## Graphing Sine and Cosine Using Amplitude and Period

15 minutes

The final portion includes a sine and a cosine function to graph, both with a period and amplitude stretch. I have the students graph these and then model them myself on the board.  This is the opportunity to show students what level of accuracy and what specific detail I would like them to include in a graph.  I add examples if time allows and it seems like the students need it.

## Homework

2 minutes

There are four portions to this Homework.  The first portion asks the students to identify the period and amplitude from the graph of a sine or cosine function.  The section portion asks them to find the same thing from an equation.  Next, they are asked to graph some sine and cosine functions.  This is the most important part.  There is no better way to memorize the important features of a trig graph than draw it given different variations.  Half of the problems are in degree and half are in radians.  The final question is an extension question asking the students to compare the domain and range of the trig graphs they just completed (Math Practice 2).

## Exit Ticket

2 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

This Exit Ticket checks that the students understand the where to find the period and amplitude in a trigonometric function.