The goal of the activity is for students to deepen their understanding about the common misconception that translating a graph to the left means subtracting from the input. Students then use function notation to write an equation for a new function that is translated both vertically and horizontally from the original function.

Listen for students who have different explanations for how to help Andre, such as:

• Rephrasing the situation.
• Citing specific points on the graph.
• Comparing the graphs of , , and .

The purpose of this discussion is for students to clarify their thinking around translating a graph horizontally and representing the translated function with an equation. Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “How could you help Andre understand the error in his thinking that ?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback: 

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”
• “How do you know ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

After Stronger and Clearer Each Time, select previously identified students to share their responses to the first question. Encourage students to continue asking questions and sharing ideas until the whole class is convinced that makes sense instead of when translating the graph 2 units to the left.

The goal of this activity is for students to describe a graphical transformation of a function from the algebraic transformation. They consider two different transformations of the same exponential function, eventually graphing both on the same axis and making sense of what the models say about the situation.

Students examine a function that is used to fit a data set, then adjust the function to make it fit a data set. As they analyze the adjustments to the functions and interpret the results, students are engaging in mathematical modeling (MP4).

If students are unsure how to describe the transformation from to or to , consider saying:

• “Describe in words the transformations Jada and Noah did to the original function .”

• “How could rewriting or as functions in terms of help you?”

Select students to share how they identified Jada's and Noah's transformations from only the equations. To help students make connections between this work and earlier work with both function notation and exponential functions, here are some possible questions for discussion:

• “What type of function is ?” ( is an exponential function.)
• “If you wrote Jada's equation in terms of , what would it look like?” ( )
• “If you wrote Noah's equation in terms of , what would it look like?” ( )
• “Let's say the person who took the meat out forgets about it and leaves town for a week. What do Jada's and Noah's functions predict the temperature of the meat will be in two days? Is it reasonable?” ( is about  while is about . Both of these are reasonable since the meat should reach the temperature of the room, and both of these temperatures are normal for a room.)