# Lesson 6

Symmetry in Equations

## 6.1: Notice and Wonder: Same and Different (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that we can tell if a function is odd from its equation, which will be useful when students identify even and odd functions from equations in a later activity. While students may notice and wonder many things about these graphs and equations, the relationship between the equations and how the negatives do and do not change the equations are the important discussion points (MP7). Additionally, this warm-up provides another opportunity for students to recognize features of odd functions, which are not as apparent as those of even functions.

### Launch

Display the graphs and equations 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 and equations. 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 the exponent of each term being odd does not come up during the conversation, ask students to discuss this idea. If time allows, contrast function $$g$$ with a different polynomial function, such as $$p(x)=x^6-x^2$$, which is even. While students do not need to memorize that polynomial functions are odd when all their terms have odd degree and even when all their terms have even degree, it can be a useful way to remember the distinction between odd and even functions in general (and that lots of functions are neither).

## 6.2: Finish the Graph (15 minutes)

### Activity

Extending from their earlier work, the purpose of this activity is for students to create the graphs of even and odd functions when given half of a graph to start with.

Monitor for students who are using different methods to verify the definitions of even and odd, including:

• drawing the right side of the curve by eye
• reflecting the curve, possibly using patty paper
• plotting specific points using symmetry to "connect the dots" when they draw in the curve
• checking points using an equation (for example, knowing that in even functions $$f(x)=f(\text-x)$$ is true means $$f(\text-1.4)=f(1.4)$$, so $$f(1.4)=0.4$$)

### Launch

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because. . .”, “I noticed _____ so I. . .”, or “How did you get. . .?”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

Here is a graph of $$y = f(x)$$ for $$\text-5 \le x \le 0$$. Draw the graph for $$0 \le x \le 5$$ and be prepared to explain your reasoning if:

1. $$f$$ is even

2. $$f$$ is odd

3. $$f$$ is neither even nor odd

### Anticipated Misconceptions

Students who aren't sure where to begin may want to use tracing paper to assist in drawing the reflections.

### Activity Synthesis

Display the graphs for the even and odd functions for all to see. Select previously identified students to share how they completed their graphs in the order listed in the Activity Narrative, recording the responses for all to see.

If students do not make connections between the different ways possible to complete the graphs, invite them to do so. A key idea here is how we can think about even and odd functions in different ways, such as a specific transformation of the graph or by the symmetry of points on the curve, and reach the same answers.

Conclude the discussion by selecting students to share how they completed the graph that is neither even nor odd. An important takeaway for students is that while there is only one way to complete the graph to make an even function and one way to make an odd function, there are many ways to make a function that is neither.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. Invite students to quietly review the three completed graphs of at least two other peers in the room. Give students quiet think time to consider what is the same and what is different about the completed graphs. Next, ask students to find a partner with whom to discuss what they noticed. Wherever possible, listen for and amplify student words and actions that describe the processes for constructing the graphs and connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Cultivate conversation

## 6.3: Odd and Even Equations (15 minutes)

### Activity

In this activity, students take turns with a partner to determine if a function whose equation is given is even, odd, or neither. Students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3). While students may begin the activity by considering particular points or sketching the graph (by hand or using graphing technology), after an initial work period a discussion will direct students toward verifying functions are even, odd, or neither algebraically.

Monitor for students who use different methods in the first 5 minutes such as

• checking specific points for symmetry
• graphing then reflecting the curve to see if it results in the same curve
• checking points in an equation such as $$f(x)=f(\text-x)$$

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Arrange students in groups of 2.

After 5 minutes of work time, pause the class. Invite previously selected students to share their methods for functions $$f$$ and $$g$$. If equations are not mentioned by students, remind them that in a previous lesson it was established that if function $$f$$ is even, then we also know $$f(x)=f(\text-x)$$, and if function $$g$$ is odd, then $$g(x)=\text-g(\text-x)$$. This is a case where the reverse is also true. For example, if $$f(x)=f(\text-x)$$, then function $$f$$ is even. This means we can evaluate a function at $$\text-x$$ algebraically to determine if a function is odd or even. For example,

1. \begin{align} f(x) &= 3x^4 - 2x^2 + 1 \\ f(\text-x) &= 3(\text-x)^4 - 2(\text-x)^2 + 1 \\ f(\text-x) &= 3x^4 - 2x^2 + 1 \\ f(\text-x)&=f(x) \end{align}
So $$f(x)$$ is even.

2. \begin{align} g(x) &= x^3 - x \\ g(\text-x) &= (\text-x)^3-(\text-x) \\ g(\text-x) &= \text-x^3+x \\ \text-g(\text-x) &= x^3-x \\ \text-g(\text-x) &=g(x) \end{align}
So $$g(x)$$ is odd.

Tell students to solve algebraically on their turn for the remaining questions, but their partner may check their work using an alternate method.

Speaking: MLR8 Discussion Supports. To support students in producing statements about features of the functions that are even, odd, or neither, provide sentence frames for students to use when they share their reasoning, such as: “I know _____ because….”, “The function is _____ because….”, and “Another way to show _____ is….”
Design Principle(s): Support sense-making; Optimize output (for justification)
Representation: Develop Language and Symbols. Create a chart with student examples of the different methods used to determine if a function is even, odd, or neither. Highlight the different methods students used to determine the type of function. Encourage students to refer back to the chart as they continue working with their partner.
Supports accessibility for: Conceptual processing; Memory

### Student Facing

Take turns with your partner to decide if the function is even, odd, or neither. If it’s your turn, explain to your partner how you decided. If it’s your partner’s turn, listen carefully to their reasons and decide if you agree. If you disagree, discuss your thinking and work to reach an agreement.

1. $$f(x) = 3x^4 - 2x^2 + 1$$
2. $$g(x) = x^3 - x$$
3. $$h(x) = (x^2 -1)(x^2 - 4)$$
4. $$j(x) = 2^x + 2^{\text-x}$$
5. $$k(x) = (x^3 - 1)x$$
6. $$m(x) = (x-0.9)x(x+1.1)$$
7. $$n(x) = x(x^2 -1)(x^2 - 4)$$
8. $$p(x) = (x^2 + 4)(x^2 - 3)$$
9. $$q(x) = \frac{1}{x} + x$$
10. $$r(x) = \frac{1}{x} - x$$

### Student Facing

#### Are you ready for more?

Write three equations with at least three terms each where one represents an even function, one an odd function, and one is neither even nor odd. Swap equations with your partner and identify which equations represent which type of function.

### Activity Synthesis

Much discussion takes place between partners. Invite students to share how they identified a function as even, odd, or neither, focusing on functions $$m$$, $$n$$, and $$p$$. Here are some questions for discussion:

• “Is showing that $$m(2)\neq m(\text-2)$$ enough to prove $$m$$ is not even?” (Yes, because if $$m$$ is even then $$m(x)=m(\text-x)$$ is true for all possible inputs $$x$$.)
• “Is showing that $$n(2)=n(\text-2)$$ enough to prove $$n$$ is even?” (No, a function is only even if that is true for all possible inputs, not just a specific input. $$n$$ is actually an odd function because $$n(x)=\text-n(\text-x)$$ is true for all possible inputs $$x$$.)

## Lesson Synthesis

### Lesson Synthesis

Arrange students in groups of 2–3. Ask them to think back to some of the functions they have studied this year, such as polynomials, radicals, and exponentials. Tell groups to pick a type of function they have studied and either:

1. Identify an equation for that type of function that is even.
2. Identify an equation for that type of function that is odd.
3. Explain why that type of function cannot be even or odd.

After work time, select groups to share their function and reasoning. If time allows, have groups create short presentations for their function with examples that include equations and graphs to illustrate their reasoning.

## Student Lesson Summary

### Student Facing

Remember the even function $$f$$ with this graph from earlier?

An equation for $$f$$ is $$f(x)=3x^2+1$$. Since we already know $$f$$ is even, we also know that the output at $$x$$ and $$\text-x$$ is the same for any value of $$x$$ in the domain of $$f$$. Said another way, $$f(x)=f(\text-x)$$ for all inputs $$x$$. If we didn't know $$f$$ was even, we could check by using $$\text-x$$ as the input.

For example, since $$f(x) = 3x^2 + 1$$,

\displaystyle \begin{align} f(\text-x) &= 3(\text-x)^2 + 1 \\ f(\text-x) &= 3x^2 + 1 \\ f(\text-x)&= f(x) \end{align}

which shows the function $$f$$ is even.

Let's look at a different function. Consider the function $$g$$ defined as $$g(x) = e^x - e^{\text-x}$$. Using $$\text-x$$ as the input, we have:

\displaystyle \begin{align} g(\text-x) &= e^{\text-x} - e^{\text-(\text-x)} \\ g(\text-x)&= e^{\text-x}- e^{x} \\ g(\text-x)&= \text-e^x+e^{\text-x}\\ \text-g(\text-x)&=e^x-e^{\text-x} \\ \text-g(\text-x)&=g(x) \end{align}

This means $$g$$ is odd since $$g(x) = \text-g(\text-x)$$.