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Unit 6, Lesson 18

Comparing Transformations

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Integrated Math 3

18.1

Warm-up

Standards Alignment

Building On

Instructional Routines

None

Materials

None

Activity Narrative

The purpose of this activity is for students to consider the graphs of three familiar types of functions, radical, linear, and quadratic, and to understand that when the same transformations are applied to different function types, the graphs change in the same way. This Warm-up also directly addresses that the order in which transformations are applied can matter, something which that comes up repeatedly throughout the lesson.

Launch

Arrange students in groups of 2. Give students brief quiet work time followed by sharing their thinking with a partner.

Activity

None

For each pair of graphs, be prepared to describe a transformation from the graph on top to the graph on the bottom.

<p>graph lower case f. second graph of upper case f is graph of lowercase f reflected over x axis, translated 1 unit left and 3 units down. </p>

<p>graph lowercase h of y = x squared minus 4. graph of uppercase H is lowercase h graph reflected over x axis, translated 1 unit left and 3 units down. </p>

<p>graph lower case g of y = the fraction 1 over 2 x plus 1. graph upper case G is lower case g reflected over x axis, translated 1 unit left and 3 units down. </p>

Student Response

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Building on Student Thinking

If students have difficulty identifying the transformations with precision, consider saying:

“Describe the transformations you see in your own words.”
“Mark corresponding points on the two graphs. How could those two points help you determine the transformations?”

Activity Synthesis

There are many possible transformations that take the upper graph to the lower graph. Begin the discussion by displaying the three graphs and selecting students to share their transformation steps.

If the same set of steps is suggested for all three transformations, draw students' attention to them. Otherwise ask “What happens if we reflect the upper graph across the -axis, and then translate it left 1 and down 3?” (That transformation works for all three.) Next, ask students if the transformation would work if we did the steps in a different order. (No, translating down and then reflecting across the -axis has a different result than reflecting across the -axis and then translating down.)

18.2

Activity

Instructional Routines

MLR4: Information Gap Cards

Materials

To Copy (from Blackline Masters)

What's the Transformation Cards

To Gather

Graph paper

Activity Narrative

This activity gives students an opportunity to determine and request the information needed to find and graph the transformation of a given function. In each case, the given function is a trigonometric function that has been translated horizontally or vertically and also reflected or scaled. The problem card gives the graph and equation of the trigonometric function. The student with the problem card figures out which transformations to apply to the graph and equation. The data cards give multiple ways for students to figure out the transformation. Monitor for students who:

Ask for the explicit transformations and the order in which they are applied.
Ask for the amplitude, midline, and period of the new function.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.

18.3

Activity

Optional

Instructional Routines

None

Materials

To Gather

Graphing technology

Activity Narrative

Use this activity to give students extra practice using technology to transform periodic functions.

The goal of this activity is for students to use graphing technology to transform the graph of a function to match another function. Because trigonometric functions are periodic, there are many ways to make the given transformations. Having the graphs allows students to use their visual intuition to visualize the transformations, and then the main work of the activity is translating this into function notation.

Launch

Arrange students in groups of 2. Graphing technology is needed for each group.

Action and Expression: Internalize Executive Functions. To support organization, provide students with a four-column table. Use these column headings: “vertical stretch,” “horizontal stretch,” “translate left/right,” and “translate up/down.” The table will provide visual support for students to see how the graphical transformations connect to the expressions.
Supports accessibility for: Language, Organization

Lesson Synthesis

The purpose of this discussion is for students to describe the graph of a transformation of a function in terms of the original function, similar to the work of the Warm-up.

Arrange students in groups of 2–3, and display the table for all to see. Tell groups to be prepared to describe how to transform the graph of the original function to match the new function, by only reading the equations and not doing any graphing. If time allows, have groups complete all four rows, otherwise assign groups 1–3 rows to complete.

original function	new function	describe the transformation

Select groups to share their descriptions, recording these for all to see (: translated left 1, reflected over the -axis, : translated down 1, horizontal scaling by a factor of (compressed), : vertical scaling by a factor of 3 (stretched), then translated down 1, : translated left 1 unit, then horizontal scaling by a factor of (compressed), and translated down 5). In particular, if any groups wrote an equation for the new function in terms of the original, highlight their work as one strategy for making sense of transformations. For example, . After groups give their descriptions, display the two graphs on the same axes so the class can check their thinking and revise where needed.

to access Cool-down.

Student Lesson Summary

Here are graphs of two trigonometric functions:

<p>graph of function f of x = sine of x. graph of function g of x = the fraction 1 over 2 sine left parenthesis 4 left parenthesis x - the fraction pi over 8 right parenthesis right parenthesis + 2 point 5. </p>

The function is given by . How can we transform the graph of to look like the graph of ? Looking at the graph of , we need to make the period and the amplitude smaller, translate the graph up, and translate the graph horizontally so it has a minimum at .

The amplitude of is and the period is , so we can begin by changing to . The midline of is 2.5 so we need a vertical translation of 2.5, giving us . The function has a minimum when , while has a minimum when . So a horizontal translation to the right by is needed. Putting all of this together, we have an expression for : .

Another way to think about the transformation is to first notice that has a minimum when is 0. If we translate right by , then also has a minimum at . The period of is , so we can write . The amplitude of is and it's midline is 2.5, so we end up with the expression for . This is the same as , just thinking of the horizontal translation and scaling in different orders.

Launch

Tell students that they will continue to work on identifying transformations between two functions. Display the Information Gap graphic that illustrates a framework for the routine for all to see.

Remind students of the structure of the Information Gap routine, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem and instruct them to switch roles.

Provide students with access to small sheets of graph paper.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Information Gap graphic visible throughout the activity or provide students with a physical copy.
Supports accessibility for: Memory, Organization

Activity

None

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because .”
Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.

If your teacher gives you the data card:

Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.

Student Response

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Lesson 18

Comparing Transformations

Warm-up

Standards Alignment

Building On

HSF-BF.B.3

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Addressing

Building Toward

Standards Alignment

Building On

Addressing

HSF-BF.B.3

HSF-IF.C.7.e

Building Toward

Standards Alignment

Building On

Addressing

HSF-BF.B.3

HSF-IF.C.7.e

Building Toward