# Lesson 2

Moving Functions

## 2.1: What Happened to the Equation? (5 minutes)

### Warm-up

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.

### Launch

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

### Student Facing

Graph each function using technology. Describe how to transform $$f(x)=x^2(x-2)$$ to get to the functions shown here in terms of both the graph and the equation.

1. $$h(x)=x^2(x-2)-5$$
2. $$g(x)=(x-4)^2(x-6)$$

### Activity Synthesis

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.

## 2.2: Writing Equations for Vertical Translations (15 minutes)

### Activity

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 $$h(x)$$

### Launch

Representation: Internalize Comprehension. Chunk this task into more manageable parts. Check in with students after the first 2–3 minutes of work time to look for students who notice the constant difference between the outputs of $$f(x)$$ and $$g(x)$$. Check to make sure that students are filling out the $$g(x)$$ column of the table correctly.
Supports accessibility for: Conceptual processing; Organization

### Student Facing

The graph of function $$g$$ is a vertical translation of the graph of polynomial $$f$$.

1. Complete the $$g(x)$$ column of the table.
2. If $$f(0) = \text-0.86$$, what is $$g(0)$$? Explain how you know.
3. Write an equation for $$g(x)$$ in terms of $$f(x)$$ for any input $$x$$.
4. The function $$h$$ can be written in terms of $$f$$ as $$h(x)=f(x)-2.5$$. Complete the $$h(x)$$ column of the table.

$$x$$ $$f(x)$$ $$g(x)$$ $$h(x)=f(x)-2.5$$
-4       0
-3 -5.8
-0.7 0
1.2 -3.3
2 0
5. Sketch the graph of function $$h$$.

6. Write an equation for $$g(x)$$ in terms of $$h(x)$$ for any input $$x$$.

### Activity Synthesis

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.

## 2.3: Heating the Kitchen (15 minutes)

### Activity

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.

### Launch

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.

### Student Facing

A bakery kitchen has a thermostat set to $$65^\circ \text{F}$$. Starting at 5:00 a.m., the temperature in the kitchen rises to $$85^\circ \text{F}$$ when the ovens and other kitchen equipment are turned on to bake the daily breads and pastries. The ovens are turned off at 10:00 a.m. when the baking finishes.

1. Sketch a graph of the function $$H$$ that gives the temperature in the kitchen $$H(x)$$, in degrees Fahrenheit, $$x$$ hours after midnight. 2. The bakery owner decides to change the shop hours to start and end 2 hours earlier. This means the daily baking schedule will also start and end two hours earlier. Sketch a graph of the new function $$G$$, which gives the temperature in the kitchen as a function of time. 3. Explain what $$H(10.25) = 80$$ means in this situation. Why is this reasonable?
4. If $$H(10.25) = 80$$, then what would the corresponding point on the graph of $$G$$ be? Use function notation to describe the point on the graph of $$G$$.
5. Write an equation for $$G$$ in terms of $$H$$. Explain why your equation makes sense.

### Student Facing

#### Are you ready for more?

Write an equation that defines your piecewise function, $$H$$, algebraically.

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their graphs of the temperature in the kitchen, present a discontinuous graph. For example, display a graph with a stable temperature at $$65^\circ\text{F}$$ and then a jump to $$85^\circ\text{F}$$. The graph can also include an incorrect horizontal translation. Ask students in pairs to identify the error, critique the reasoning, and write a response to the author explaining the error in reasoning. As students discuss with their partner, listen for students who justify why a sudden jump in temperature is not possible or why the horizontal translation is incorrect. Invite students to share their critiques with the class. Listen for and amplify the language students use to describe what happens to the temperature in the kitchen when the ovens are turned on, and explain why a discontinuous graph or incorrect horizontal translation is not representative of the situation. This will help students understand graphical representations and horizontal translations.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

Arrange students in groups of 2–4 and provide access to graph paper. Display the two graphs and the following prompts:

1. Keeping your work hidden from your group, draw a transformation of the graph of $$f$$ and name it $$g$$.
2. Write an equation for $$g(x)$$ in terms of $$f(x)$$.

Tell the groups to take turns having one person in the group share their equation for the other students in the group to sketch before comparing graphs. If students disagree, encourage them to discuss their thinking and work to reach agreement. If time is limited, only have one student share their equation in each group.

## Student Lesson Summary

### Student Facing

A pumpkin catapult is used to launch a pumpkin vertically into the air. The function $$h$$ gives the height $$h(t)$$, in feet, of this pumpkin above the ground $$t$$ seconds after launch.

Now consider what happens if the pumpkin had been launched at the same time, but from a platform 30 feet above the ground. Let function $$g$$ represent the height $$g(t)$$, in feet, of this pumpkin. How would the graphs of $$h$$ and $$g$$ compare?

Since the height of the second pumpkin is 30 feet greater than the first pumpkin at all times $$t$$, the graph of function $$g$$ is translated up 30 feet from the graph of function $$h$$. For example, the point $$(2,66)$$ on the graph of $$h$$ tells us that $$h(2) = 66$$, so the original pumpkin was 66 feet high after 2 seconds. The new pumpkin would be 30 feet higher than that, so $$g(2) = 96$$. Since all the outputs of $$g$$ are 30 more than the corresponding outputs of $$h$$, we can express $$g(t)$$ in terms of $$h(t)$$ using function notation as $$g(t) = h(t) + 30$$.
Now suppose instead the pumpkin launched 5 seconds later. Let function $$k$$ represent the height $$k(t)$$, in feet of this pumpkin. The graph of $$k$$ is translated right 5 seconds from the graph of $$h$$. We can also say that the output values of $$k$$ are the same as the output values of $$h$$ 5 seconds earlier. For example, $$k(7) = 66$$ and $$h(7-5) = h(2) = 66$$. This means we can express $$k(t)$$ in terms of $$h(t)$$ as $$k(t)=h(t-5).$$