### Summary

Students will use a graphing calculator to explore function transformations on polynomial (quadratic and cubic), radical (square root and cube root), and transcendental (exponential and logarithmic) functions. Students then generalize these transformations using function notation.

### Essential Question(s)

How does changing the equation of a function change its graph? How can we generalize these changes?

### Snapshot

**Engage**

Students are introduced to function transformations through a class discussion about functions with which they are already familiar.

**Explore**

Students explore function transformations through a guided investigation with a graphing utility on several parent graphs.

**Explain**

Students generalize transformations using function notation.

**Extend**

Students further demonstrate their understanding of transformations in a class presentation.

**Evaluate**

Students demonstrate their understanding by writing about how changing the equations changes the graph.

### Materials

Lesson Slides (attached)

Graphing A handout (attached; one per pair; printed front/back)

Graphing B handout (attached; one per pair; printed front/back)

Graphing C handout (attached; one per pair; printed front/back)

Put It All Together handout (attached; one per pair; printed front/back)

Chart or poster paper

Sticky Notes (two colors; one of each color per student)

Pencils

Paper

Graphing calculator (or other graphing utility)

### Engage

10 Minute(s)

Introduce the lesson using the attached **Lesson Slides**. **Slide 3** displays the lesson’s essential questions: *How does changing the equation of a function change its graph? How can we generalize these changes?* **Slide 4** identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.

Display **slide 5** and present for the students the prompt: *What do the graphs of y = x + 3 and y = |x| + 3 have in common?* Allow students time to answer for themselves and then, using the Elbow Partner strategy, have students share with an elbow partner. Take volunteers for responses.

Show **slide 6** and provide the example: *y* = 4*x* and *y* = |4*x*|. Then ask, *How are these equations similar?* Allow for discussion before asking a few students to share.

Summarize their observations.

### Explore

105 Minute(s)

Students need a graphing calculator or access to the Desmos Studio graphing calculator to complete the activity. Display **slide 7** and give each pair a copy of the attached **Graphing A** handout and a graphing utility. Here students are asked to explore the graphs of the parent graph and a few transformations for polynomials: both quadratic and cubic functions.

In Part I, students should make predictions about graphing and compare *y* = *x*^{2}, *y* = (*x* – 3)^{2}, *y* = *x*^{2} + 3, and *y* = 2*x*^{2}. Students do this again in Part II, but with cubic functions: *y* = *x*^{3}, *y* = (*x* – 3)^{3}, *y* = *x*^{3} + 3, and *y* = 2x^{3}.

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

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

Allow students 30-35 minutes to work through the activity. Walk around and guide students who are lost but try not to lead them outright. Some good questions to ask are:

What does your partner think?

How did you come to that conclusion?

Can you show me that it works in [this] case? (try a negative number or a fraction)

Why do you think that?

What have you tried already?

At the end of the class/work period, discuss the students' results.

Repeat the same process over the next couple of class/work periods with **slide 8** and the attached **Graphing B** handout for radical functions and again with **slide 9** and the attached **Graphing C** handout for exponential and logarithmic functions.

### Explain

45 Minute(s)

Show **slide 10** and split students into new groups of 2-3. Provide each group with a copy of the attached **Put It All Together** handout. Students will need access to the graphing utility again. Let students know that they are going to be generalizing their observations. Use **slide 11** for a short explanation of the usage of function notation, if needed, before students start working.

Display **slide 12** and allow students time to work through the handout, and then come back together as a class. Help them recall their previous observations and compare/contrast them with the observations they have just made.

Summarize the generalized transformations together as a class and record them in a visible place.

Once students are confident in how to represent their observations with function notation and how different constants and coefficients affect the graph of a function, move to **slide 13**. Direct students’ attention to the *Prediction* portion of their handout and ask the class to make a prediction about how y = 3·sin(*x* + 4) – 2 compares to *y* = sin(*x*).

After a few minutes, transition to **slide 14** and have students graph the two functions in their graphing calculator and compare their results with their predictions. Ask for volunteers to share their thinking to share how they were able to accurately predict how the graphs would differ.

### Extend

40 Minute(s)

Have students get into groups of 3-4 or assign groups then display **slide 15**. Assign each group one type of transformation: *f*(*x*) + *c*, *f*(*x* + *c*), *f*(*c*·*x*), and *c*·*f*(*x*). Have each group create an Anchor Chart with everything they know about their assigned transformation.

If you have more than four groups, consider either giving groups the same transformations (i.e. two groups might have *f*(*x*) + *c*) or giving some groups the challenge of multiple transformations (i.e. groups might have *a*·*f*(*x* + *c*) or *f*(*b*·*x*) + *d*). If you have fewer than four groups, consider making one of the Anchor Charts yourself. Use the hidden **slide 16** for sample student responses. Give students approximately 15 minutes to create their poster.

Show **slide 17** and introduce students to the Gallery Walk strategy. Give each student two different colored sticky notes. Indicate to students which color indicates “This transformation makes the * most* sense,” and which color indicates “This transformation makes the

*sense.” Emphasize to students that they are not evaluating the quality of the Anchor Chart but are expressing their level of understanding. Explain to students that they are to read each poster then put their sticky notes on the corresponding posters.*

**least**### Evaluate

5 Minute(s)

Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Display **slide 18** and provide the prompt on the slide asking students to explain how *a*, *b*, *c*, and *d* in the equation *y* = *a*·*f*(*b*·*x* + *c*) + *d* affect the parent graph. Allow students a couple of minutes to respond using an index card, sticky note, piece of paper, etc. Use these responses as a formative assessment to see students’ understanding of this concept.

### Resources

IncptMobis. (2024).

*Graphing Calculator X84*(Version 3.2) [Mobile app]. App Store. https://iphonecalculator.com/K20 Center. (n.d.). Anchor Chart. Strategies. https://learn.k20center.ou.edu/strategy/58

K20 Center. (n.d.). Bell Ringers and Exit Tickets. Strategies. https://learn.k20center.ou.edu/strategy/125

K20 Center. (n.d.). Desmos Studio. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2356

K20 Center. (n.d.). Elbow Partners. Strategies. https://learn.k20center.ou.edu/strategy/116

K20 Center. (n.d.). Fist to Five. Strategies. https://learn.k20center.ou.edu/strategy/68

K20 Center. (n.d.). Gallery Walk / Carousel. Strategies. https://learn.k20center.ou.edu/strategy/118

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