Lesson 2

The purpose of this warm-up is for students to consider both graphs and equations of functions when describing horizontal and vertical translations. This work builds on the more informal descriptions of the previous lesson and looks ahead to following activities where students describe transformations using function notation. 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 \(x\). Students will continue to work on this skill throughout the unit, so this activity is only meant as an introduction.

Provide access to devices that can run Desmos or other graphing technology.

Display the graphs of the three functions for all to see. Begin the discussion by inviting students to share their observations of the changes in both the equations and graphs, recording these next to the relevant graph.

An important takeaway for students from this discussion is understanding that we can define equations for \(g(x)\) and \(h(x)\) in terms of \(f(x)\). Build on student descriptions, such as “\(h(x)\) is like \(f(x)\), but the output has a -5” or “\(g(x)\) is like \(f(x)\) but all the inputs have a -4,” to help students see how we can write \(h(x)=f(x)-5\) and \(g(x)=f(x-4)\). Students will develop their ability to write equations such as these throughout the rest of the lesson and in several future lessons, so they are not expected to have mastery at this time.

Continuing the thinking from the warm-up, students now compare three functions whose graphs are vertical translations from one another and generalize their observations of the graphs and data tables into equations where one function is defined in terms of another.

Monitor for students making connections between representations as they write an equation for \(g(x)\) in terms of \(h(x)\). For example, students may:

• compare the graphs of \(g\) and \(h\), 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 \(g(x)\) in terms of \(f(x)\) and \(h(x)=f(x)-2.5\) to rewrite \(g(x)\) in terms of \(h(x)\)

A key takeaway from this discussion is that students understand that a vertical translation from one graph to another corresponds to constant differences between the outputs of the functions for the same input. We can then generalize these point by point comparisons using function notation, where adding or subtracting from the output of the function resulting in a vertical translation of its graph.

Select previously identified students to share the connections they made between the representations in the order listed in the Activity Narrative. If time allows and no students used the equations for \(g(x)\) and \(h(x)\) in terms of \(f(x)\) to write a new equation for \(g(x)\) in terms of \(h(x)\), invite them to do so now.

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.

If necessary, it may be helpful to tell students that this is a simplified situation. In an actual bakery, temperatures in the kitchen are more likely to vary instead of staying at an exact degree.

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.

Display the previously selected graphs for all to see. 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 \(H(10.25)\) and the corresponding point on the graph of \(G\). Connect the horizontal translation of each point to changing the input of the function. Next, invite students to share their equation for \(G\) in terms of \(H\) and how they reasoned that the input of \(x+2\) for \(H\) corresponds to the input of \(x\) for \(G\). Unlike vertical translations, horizontal translations can seem to work backwards. Because of this, it is important to give students time to make sense of different explanations for why the graph of \(G\), when translated 2 hours to the left of the graph of \(H\), means \(G(x)=H(x+2)\). Students will continue making sense of horizontal translations in the following lesson.