The purpose of this Warm-up is for students to consider graphs and equations of functions when describing horizontal and vertical translations. Students are building from the more informal descriptions that they used as they increase the precision of their descriptions using information from the graphs and equations (MP6). 

During the whole-class discussion, students see that when functions are transformations of one another, we can use equations to define one function in terms of the other for any input . Students will continue to work on this skill throughout the unit, so this activity is meant as an introduction to the idea.

The purpose of this discussion is for students to familiarize themselves with representing one function in terms of another function. Display the graphs of the three functions. Invite students to share their observations of the differences between the equations and graphs of each function, recording these observations next to the relevant graph.

An important takeaway for students from this discussion is understanding that we can define equations for and in terms of . Build on student descriptions to demonstrate how to write the explanation using function notation.

• If a student says, “ is like , but the output has a -5,” display .
• If a student says, “ is like , but all the inputs have a -4,” display .

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 .

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.