# Lesson 3

More Movement

## 3.1: Moving a Graph (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that a graph can be translated horizontally and vertically on the coordinate plane in a variety of ways. One graph has a different shape than the center graph, \(A\). This is purposeful to make the point that horizontal and vertical translations only change the location of a graph and not the shape. In a later lesson, using scale factors to squash or stretch a graph will be considered. Another important takeaway is that it is sometimes possible to describe a sequences of translations (sometimes called a transformation or rigid transformation) in different correct ways. In this case, whether the horizontal or vertical translation comes first does not change the outcome.

### Launch

Arrange students in groups of 2. Display the graph for all to see. Tell partners that their job is to be prepared to explain how to transform the graph of \(A\) to look like any of the other graphs using horizontal and vertical translations. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

### Student Facing

How can we translate the graph of \(A\) to match one of the other graphs?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Invite students to share how they see the translation(s) need to take \(A\) to one of the other graphs. If graph \(D\) being a different shape than the other graphs does not come up during the conversation, ask students to discuss this idea with particular emphasis that since the shape does not match, there is no combination of vertical and horizontal translations that can take the graph of \(A\) to \(D\).

Conclude the discussion by telling students that a sequence of translations is often called a *transformation* or, more specifically, a *rigid transformation* (since translations don't change the shape of the graph). In this warm-up, we described the transformations that take the graph of \(A\) to the graphs of \(B\), \(C\), \(E\), and \(F\).

## 3.2: New Hours for the Kitchen (15 minutes)

### Activity

Working with a context from a previous lesson, 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.

Monitor for students who have different explanations for how to help Andre, such as by rephrasing the situation or citing specific points on the graph, to share during the whole-class discussion.

### Launch

As with the previous time this situation was discussed, it may be helpful to tell students that this is a simplified situation. In an actual bakery, temperatures in the kitchen likely vary instead of staying a set temperature.

*Representation: Internalize Comprehension.*Begin the activity with familiar representations. Some students may benefit from creating a 3-column table to compare outputs for the temperatures of the bakery for the original hours and when it opens 2 hours earlier. Remind students to use the same times (hours after midnight) for the input column.

*Supports accessibility for: Conceptual processing; Memory*

### Student Facing

Remember the bakery with the thermostat set to \(65^\circ \text{F}\)? At 5:00 a.m., the temperature in the kitchen rises to \(85^\circ \text{F}\) due to the ovens and other kitchen equipment being used until they are turned off at 10:00 a.m. When the owner decided to open 2 hours earlier, the baking schedule changed to match.

- Andre thinks, “When the bakery starts baking 2 hours earlier, that means I need to subtract 2 from \(x\) and that \(G(x)=H(x-2)\).” How could you help Andre understand the error in his thinking?
- The graph of \(F\) shows the temperatures after a change to the thermostat settings. What did the owner do?
- Write an expression for \(F\) in terms of the original baking schedule, \(H\).

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

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 why \(G(x)=H(x+2)\) makes sense instead of \(G(x)=H(x-2)\) when translating the graph left 2 hours.

Invite students to share their equations for \(F\) in terms of \(H\). Connect the \((x-1)\) translating the graph 1 unit to the right with the explanations of \((x+2)\) translating the graph two units to the left. Encourage students to examine specific points as they test their equation.

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. As each selected student shares their response to the first question, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.

*Design Principle(s): Support sense-making*

## 3.3: Thawing Meat (15 minutes)

### Activity

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.

By examining how the equation of a function was changed to make it fit a data set, students are building skills that will help them in mathematical modeling (MP4).

### Launch

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

*Writing, Conversing: MLR1 Stronger and Clearer Each Time.*Use this routine to help students improve their writing by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 2–3 partners to share their response to the last question. Students should first check to see if they agree with each other about whether Jada’s or Noah’s graph best fits the data. Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “How do you know...?” or “Why do you think...?” Next, provide students with 3–4 minutes to revise their initial draft based on feedback from their peers. This will help students evaluate graphical and algebraic transformations and justify functions that model data.

*Design Principle(s): Support sense-making; Optimize output (for explanation)*

### Student Facing

A piece of meat is taken out of the freezer to thaw. The data points show its temperature \(M\), in degrees Fahrenheit, \(t\) hours after it was taken out. The graph \(M=G(t)\), where \(G(t) = \text-62(0.85)^t\), models the shape of the data but is in the wrong position.

\(t\) | \(M\) |
---|---|

0 | 13.1 |

0.41 | 22.9 |

1.84 | 29 |

2.37 | 36.1 |

2.95 | 36.8 |

3.53 | 38.8 |

3.74 | 40 |

4.17 | 42.2 |

4.92 | 45.8 |

Jada thinks changing the equation to \(J(t)=\text-62(0.85)^t + 75.1\) makes a good model for the data. Noah thinks \(N(t) = \text-62(0.85)^{(t+1)}+68\) is better.

- Without graphing, describe how Jada and Noah each transformed the graph of \(G\) to make their new functions to fit the data.
- Use technology to graph the data, \(J\) and \(N\), on the same axes.
- Whose function do you think best fits the data? Be prepared to explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Elena excludes the first data point and chooses a linear model, \(E(t)=21.32+5.06t\), to fit the remaining data.

- How well does Elena’s model fit the data?
- Is Elena’s idea to exclude the first data point a good one? Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students may be unsure how to describe the transformation from \(G\) to \(J\) or \(G\) to \(N\). Encourage these students to try and write \(J\) and \(N\) as functions in terms of \(G\).

### Activity Synthesis

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 \(G\)?” (\(G\) is an exponential function.)
- “If you wrote Jada's equation in terms of \(G\), what would it look like?” (\(J(t) = G(t)+75.1\).)
- “If you wrote Noah's equation in terms of \(G\), what would it look like?” (\(N(t) = G(t+1)+68\).)
- “Let's say the person who took the meat out forgot 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?” (\(J(48)\) is about \(75^\circ \text{F}\) while \(N(48)\) is about \(68^\circ \text{F}\). 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.)

## Lesson Synthesis

### Lesson Synthesis

Display the image from the warm-up.

Tell students they are going to pick three of the functions and write equations for them in terms of one of the other functions. For example, \(C(x)=B(x+9)-5\). After 2 minutes of quiet work time, invite students to share their equations, recording responses for all to see and giving time for students to check each other's thinking.

## 3.4: Cool-down - Translate This (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

Remember the pumpkin catapult? The function \(h\) gives the height \(h(t)\), in feet, of the pumpkin above the ground \(t\) seconds after launch. Now suppose \(k\) represents the height \(k(t)\), in feet, of the pumpkin if it were launched 5 seconds later. If we graph \(k\) and \(h\) on the same axes they looks identical, but the graph of \(k\) is translated 5 units to the right of the graph of \(g\).

Since we know the pumpkin's height \(k(t)\) at time \(t\) is the same as the height \(h(t)\) of the original pumpkin at time \(t-5\), we can write \(k\) in terms of \(h\) as \(k(t)=h(t-5)\).

Suppose there was a third function, \(j\), where \(j(t)=h(t+4)\). Even without graphing \(j\), we know that the graph reaches a maximum height of 66 feet. To evaluate \(j(t)\) we evaluate \(h\) at the input \(t+4\), which is zero when \(t = \text-4\). So the graph of \(j\) is translated 4 seconds to the left of the graph of \(h\). This means that \(j(t)\) is the height, in feet, of a pumpkin launched from the catapult 4 seconds earlier.