# Lesson 4

Reflecting Functions

## 4.1: Notice and Wonder: Reflections (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that graphs can be reflected both vertically and horizontally, which will be useful when students generate a table and graph for $$\text-f(x)$$ and $$f(\text-x)$$ from $$f(x)$$ in a later activity. Students should be familiar with the properties of reflections from a previous course. While students may notice and wonder many things about these images, being precise about how the graphs are related is the important discussion point.

This prompt gives students opportunities to see and make use of structure (MP7). They should notice that the distance of each point on the graph to the line of reflection is the same but in a different direction after the graph is reflected. In this unit, reflections across the $$x$$- and $$y$$-axes are the focus.

### Launch

Display the three graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the graphs. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If lines of reflection do not come up during the conversation, ask students to discuss this idea. Students will focus on what values do and do not change when reflecting across the $$x$$- and $$y$$-axes during this lesson, so it is okay if students do not focus on specific coordinate values at this time.

## 4.2: Reflecting Across (15 minutes)

### Activity

Building on the warm-up, in this activity students compare two functions where one is a reflection of the other across the $$x$$-axis. They first use coordinates on the graph of $$f$$ to complete a table and compute $$g(x)=\text-f(x)$$ for each $$x$$-value in the table. Students then plot $$g$$ on the same axes as $$f$$, allowing them to directly compare the graphs of $$f$$ and $$g$$ before explaining why $$g(x) = \text-f(x)$$ means the graph of $$g$$ is a reflection of the graph of $$f$$ across the $$x$$-axis.

### Launch

Display the graph of function, $$f$$, from the Task Statement and explain that a different function, $$g$$, is defined by $$g(x)=\text-f(x)$$. Ask students, “What is $$f(\text-1)$$? $$g(\text-1)$$?” After some quiet think time, invite students to share their responses and explain how they determined their answer. Make sure the discussion includes the key idea that the point $$(\text-1,\text-4)$$ is on the graph of $$f$$ and the point $$(\text-1,4)$$ is on the graph of $$g$$.

Speaking: MLR8 Discussion Supports. To support students in producing statements about their reasoning for what $$f(\text-1)$$ and $$g(\text-1)$$ are in the launch, provide sentence frames for students to use. For example, “I noticed _____ , so I...” and “I know _____ because….”
Design Principle(s): Support sense-making; Optimize output (for comparison)

### Student Facing

Here is the graph of function $$f$$ and a table of values.

$$x$$ $$f(x)$$ $$g(x) = \text-f(x)$$
-3 0
-1.5 -4.3
-1 -4
0 -1.8
0.6 0
2.6 3.9
4 0
1. Let $$g$$ be the function defined by $$g(x) = \text-f(x)$$. Complete the table.
2. Sketch the graph of $$g$$ on the same axes as the graph of $$f$$ but in a different color.
3. Describe how to transform the graph of $$f$$ into the graph of $$g$$. Explain how the equation produces this transformation.

### Anticipated Misconceptions

Students may think that the two graphs are related by a different type of transformation: reflection over the $$y$$-axis and 180 degree rotation around $$(0.6,0)$$ are both reasonable guesses. Have these students (and any other students questioning if their answer is correct) use tracing paper to check their initial guesses.

### Activity Synthesis

Invite students to share their explanation of how the graph of $$g$$ is related to the graph of $$f$$ and why. The key idea in this discussion is that $$g(x) = \text-f(x)$$ produces a reflection across the $$x$$-axis by holding the inputs constant and taking the outputs to their opposites. Students may recall from geometry that in a reflection, the line of symmetry (the $$x$$-axis here) makes a perpendicular bisector when connecting a point to its image, but this level of detail is not necessary.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, invite students to identify points from the table to the graph by using a different color and labeling the ordered pairs on the graph.
Supports accessibility for: Visual-spatial processing

## 4.3: Reflecting Across a Different Way (15 minutes)

### Activity

Continuing the thinking from the previous activity with the same starting function, students now compare two functions where one is a reflection of the other across the $$y$$-axis. Keeping the same activity structure, they first use the coordinates on the graph of $$f$$ to complete a table and compute $$h(x) = f(\text-x)$$ for corresponding points in the table. Unlike the previous activity, students fill in both $$x$$- and $$y$$-coordinates for the function $$h$$. They then plot $$h$$ on the same axes as $$f$$ to compare the graphs directly before explaining why $$h(x) = f(\text-x)$$ means the graph of $$h$$ is a reflection of the graph of $$f$$ across the $$y$$-axis.

### Launch

Arrange students in groups of 2. Display the graph of function $$f$$ and the prompt:

• Let $$h$$ be the function defined by $$h(x) = f(\text-x)$$. Make a prediction about what the graph of $$h$$ will look like.

Allow students 1 minute of quiet think time, then invite students to briefly discuss their predictions with their partner. Tell students that they will confirm their prediction in the activity.

### Student Facing

Here is another copy of the graph of $$f$$ from the earlier activity. This time, let $$h$$ be the function defined by $$h(x) = f(\text-x)$$.

1. Use the definition of $$h$$ to find $$h(0)$$. Does your answer agree with your prediction?
2. What does your prediction tell you about $$h(\text-0.6)$$? Does your answer agree with the definition of $$h$$?
3. Complete the tables. The values for $$x$$ will not be the same for the two tables.

$$x$$ $$f(x)$$
-3 0
-1.5 -4.3
-1 -4
0 -1.8
0.6 0
2.6 3.9
4 0
$$x$$ $$h(x)=f(\text-x)$$

4. Sketch the graph of $$h$$ on the same axes as the graph of $$f$$ but in a different color.
5. Describe what happened to the graph of $$f$$ to transform it into the graph of $$h$$. Explain how the equation produces this transformation.

### Student Facing

#### Are you ready for more?

1. Describe how the graph of $$h$$ relates to the graph of $$g$$ defined in the earlier activity.
2. Write an equation relating $$h$$ and $$g$$.

### Anticipated Misconceptions

Students may think that the two graphs are related by a different type of transformation: reflection over the $$x$$-axis and 180 degree rotation around $$(0.6,0)$$ are both reasonable guesses. Have these students (and any other students questioning if their answer is correct) use tracing paper to check their initial guesses.

### Activity Synthesis

Invite students to share their explanation of how the graph of $$h$$ is related to the graph of $$f$$ and why. The key idea in this discussion is that $$h(x) = f(\text-x)$$ produces a reflection across the $$y$$-axis by holding the output values constant and taking the input values to their opposites.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, invite students to identify points from the table to the graph by using a different color and labeling the ordered pairs on the graph.
Supports accessibility for: Visual-spatial processing; Organization

## Lesson Synthesis

### Lesson Synthesis

Display the image and prompt for all to see:

Describe how to transform the graph of $$f$$ to get the graph of:

1. $$p(x)= f(\text-x) + 4$$
2. $$q(x)=\text-f(x - 7)$$
3. $$r(x)=\text-f(\text-x)$$

After some quiet think time, invite students to share their responses. Sample responses:

1. Reflect across the $$y$$-axis and translate up 4 units.
2. Translate to the right 7 units and then reflect across the $$x$$-axis.
3. Reflect across the $$y$$-axis and then reflect across the $$x$$-axis.

If time allows, highlight for students that for these 3 graphs, the order in which they complete the transformations doesn't matter. For example, if we reflect $$f$$ across the $$y$$-axis and then translate up 4 units or we translate the graph of $$f$$ up 4 units and then reflect it across the $$y$$-axis, we get the same graph. Invite students to think of a transformation with at least two moves where the order does matter. (Translating up 2 and reflecting across the $$x$$-axis is different from reflecting across the $$x$$-axis and then translating up 2.)

## 4.4: Cool-down - Two Reflections (5 minutes)

### Cool-Down

Here are graphs of the functions $$f$$, $$g$$, and $$h$$, where $$g(x)=\text-f(x)$$ and $$h(x)=f(\text-x)$$. How do these equations match the transformation we see from $$f$$ to $$g$$ and from $$f$$ to $$h$$?
Considering first the equation $$g(x)=\text-f(x)$$, we know that for the same input $$x$$, the value of $$g(x)$$ will be the opposite of the value of $$f(x)$$. For example, since $$f(0)=1$$, we know that $$g(0)=\text-f(0)=\text-1$$. We can see this relationship in the graphs where $$g$$ is the reflection of $$f$$ across the $$x$$-axis.
Looking at $$h(x)=f(\text-x)$$, this equation tells us that the two functions have the same output for opposite inputs. For example, 1 and -1 are opposites, so $$h(1)=f(\text-1)$$ (and $$h(\text-1)=f(1)$$ is also true!). We can see this relationship in the graphs where $$h$$ is the reflection of $$f$$ across the $$y$$-axis.