In this activity, students compare three functions whose graphs are vertical translations from one another. Students use repeated reasoning as they generalize their observations of the graphs and data tables into equations where one function is defined in terms of another (MP8).

Monitor for students making connections between representations as they write an equation for in terms of . For example, students may:

• Compare the graphs of and , perhaps sketching them on the same axes and then measuring the vertical distance between them.
• Calculate the difference between the outputs from the table.
• Combine the equation they wrote for in terms of and to rewrite in terms of .

The goal of this discussion is for students to compare, contrast, and connect the different representations of one function in terms of another. 

Display 2–3 approaches from previously selected students. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “How are the table, graph, and equations the same? How are they different?”
• “How does the translation of from show up in each representation?” (On the graph, we can see that is translated up by 6.5 units. On the table, we can see that the difference between and is 6.5 for each point. With the equation, we can see the constant term of 6.5, so is always 6.5 more than for any input.)
• “Are there any benefits or drawbacks to one representation compared to another?”

If no students used the equations for and in terms of to write a new equation for in terms of , display a representation like this one:

Ask students to describe how they could use this strategy to define  in terms of .

This activity changes the focus from vertical to horizontal translations. Students first graph the relationships between time and temperature for two situations related by a horizontal transformation. They then make sense of specific points on the graphs before generalizing the relationship between the two situations, using function notation to capture the horizontal translation.

Monitor for students who sketch graphs of different steepness or who make discontinuous graphs for comparison during the whole-class discussion.

Make clear that we are not graphing the temperature the thermostat is set at, but rather the actual temperature inside. This means that there will be time when the temperature is increasing or decreasing between temperature settings, which appears as a diagonal line on the graph.

The purpose of this discussion is for students to make sense of how horizontal translations are represented on a graph and in an equation. Display the previously selected graphs. Invite students to discuss whether each graph is possible and what kind of conditions it represents. (For example, a jump discontinuity in a graph is not possible.)

Select students to share their explanations for the meaning of and the corresponding point on the graph of . Connect the horizontal translation of each point to changing the input of the function. Next, invite students to share their equation for in terms of and how they reasoned that the input of for corresponds to the input of for . Unlike vertical translations, horizontal translations can seem to work backward. Because of this, it is important to give students time to make sense of different explanations for why the graph of , when translated 2 hours to the left of the graph of , means . Students will continue making sense of horizontal translations in the following lesson.