Lesson 3

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.

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. 

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 \(F\).

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.

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.

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.

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).

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

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 \(G\).

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.)