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 strategies 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 is true means , so ).

Plan to have students present in this order to support the generalization of the properties of even and odd functions.

If students are not sure how to get started, consider saying:

• “Tell me more about what you know is true for even functions.”

• “How could you use tracing paper to help you draw the reflections?”

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

Connect the different responses to the learning goals by asking questions such as:

• “How did these different approaches result in the same graph?”
• “What generalizations can we make based on these examples?”

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 even nor odd.

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 that 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 .

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

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

• “What were some ways you handled functions that were neither even nor odd?”
• “Describe any difficulties you experienced and how you resolved them.”
• “Is showing that   enough to prove is not even?” (Yes, because if is even then  is true for all possible inputs .)
• “Is showing that  enough to prove is even?” (No, a function is even only if that is true for all possible inputs, not for just a specific input. is actually an odd function because is true for all possible inputs .)