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 many similar or different 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 as students consider the structure of the functions (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.

Ask students to share the things that were similar and different. 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 additional that you notice 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 with a different polynomial function, such as , 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).

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.

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).