If students are not sure where to start with the graphs, consider saying:

• “Tell me more about what you expect the graph of to look like compared to the graph of .”

• “How could making a table help you sketch a graph of ?”

The purpose of this discussion is for students to articulate how and are transformations of .

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:
  “I can find by taking half the difference of and and subtracting it from 212. So is 196.5.” 

Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.

• Give students 2–3 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

If time allows, ask: “How are and related to ?” (For , the outputs are twice as large as . For , the inputs are half as large, so it stretches out to take twice as long for the same outputs to happen.)

In this activity, students will both write equations to represent transformations shown by graphs and graph transformations described by equations. A piecewise function is used to help students focus on how the inputs or outputs (or both) are affected by scale factors.

Encourage students to use precise language when describing the transformations and to make sure their words match the transformations they describe algebraically (MP6). In particular, students need to be clear whether a scale factor affects the output or the input and if it affects the graph by moving points toward an axis (compressing) or away from an axis (stretching).

Display the graphs of , , and for all to see. Here are some questions for discussion:

• “How did you figure out the highest point on the graph of ? What was it?” (Since , has all the same outputs as , just at different inputs. This means the highest point on is , and that happens when .)
• “Name another transformation we could do to where the resulting graph would have a highest point at 2.” (Any horizontal scale factor or horizontal translation. also has a highest point at 2 since has a lowest point at -2.)

If time allows, conclude the discussion by challenging students to come up with at least two possible equations for . ( , , , and are 4 possibilities.) Invite students to share their ideas, recording responses for all to see.