### Summary

Students will use a graphing tool to analyze basic transformations on linear, absolute value, and quadratic parent graphs in order to create general rules about function transformations.

### Essential Question(s)

How are the transformations of parent graphs related?

### Snapshot

**Engage**

Students begin their exploration of function transformations by analyzing linear functions with which they should already be familiar.

**Explore**

Small groups explore function transformations through a guided investigation with a graphing utility on absolute value and quadratic functions with the goal being for them to generalize the transformation rules.

**Explain**

Students come to consensus on the general transformation rules in a class discussion.

**Extend**

Students create their own graphs and equations before switching with another group to determine the corresponding equations or graphs.

**Evaluate**

Students predict the effects of a negative leading coefficient.

### Materials

Lesson Slides (attached)

Exploring Graphs handout (attached; one per student; printed front/back)

Pencil

Paper

Graphing calculator (or other graphing utility)

### Engage

15 Minute(s)

Introduce the lesson using the attached **Lesson Slides**. **Slide 3** displays the lesson’s essential question: *How are the transformations of parent graphs related?* **Slide 4** identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.

Before starting the activity, have students pick a partner. Students need a graphing calculator or access to the Desmos Studio graphing calculator to complete the activity. Go to **slide 5** and display the equations *y* = *x*, *y* = 2*x*, and *y* = ½*x* for the students. Have the students use the graphing calculator to graph each function and ask them to compare and contrast the resulting graphs. Using Think-Pair-Share, have students share their answers. Choose a few students and ask them to share their answers with the class.

Go to **slide 6** and pose the same question using *y* = *x* + 1, *y* = *x* – 2, and *y* = *x* + 3. Discuss how these changes resulted in different graphs than the previous ones.

Discuss with students the idea of a parent graph.

### Explore

45 Minute(s)

Go to **slide 7** and have students continue to work with their partner to complete the attached **Exploring Graphs** handout. Allow students to work in their pairs independently but monitor their progress.

In Part I, students should make predictions about graphing and comparing *y* = |*x*|, *y* = |*x* – 1|, *y* = |*x *+ 3|, *y* = |*x*| – 2, and *y* = |*x –* 2| + 3.

Students do this again in Part II, but with quadratic functions: *y* = *x*², *y* = (*x* – 3)², *y* = (*x* + 1)², *y* = *x*² + 4, and *y* = (*x* – 2)² + 3.

In Part III, students are asked to compare their absolute value and quadratic graphs to list observations and patterns.

In Part IV, each group then joins another group to compare what they observed.

### Explain

25 Minute(s)

Go to **slide 8** and have students share what they observed from the Exploring Graphs handout with the class.

What were the changes to the absolute value function?

What were the changes to the quadratic function?

What are the general rules of transformations?

Make sure there is consensus on the transformation general rules.

### Extend

25 Minute(s)

Go to **slide 9** and have students create their own graphs and equations in pairs to trade with another pair. Have students work with their original partner to create:

1 absolute value equation (no graph)

1 absolute value graph (no equation)

1 quadratic equation (no graph)

1 quadratic graph (no equation)

After they have completed this task, have pairs trade their creations with another pair. Each pair now needs to find the missing graph and equation for each provided equation and graph.

### Evaluate

10 Minute(s)

Go to **slide 10**. Again, using Think-Pair-Share, have students predict how the graph of *y* = –|*x* + 3| – 5 would differ from the parent function.

After giving students time to think about the answer (and writing something down), ask for a few volunteers to share their predictions. Then move to **slide 11** and facilitate a class discussion comparing the graph of given absolute value function compares to the parent graph. Use this time to answer any clarifying questions.

### Resources

IncptMobis. (2024).

*Graphing Calculator X84*(Version 3.2) [Mobile app]. App Store. https://iphonecalculator.com/K20 Center. (n.d.). Desmos Studio. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2356

K20 Center. (n.d.). Think-Pair-Share. Strategies. https://learn.k20center.ou.edu/strategy/139

Pixabay. (December 9, 2018). Figurine [Photograph]. Pixabay. https://pixabay.com/images/id-3871893/