The purpose of this activity is for students to identify common features in graphs of functions that are the same when reflected across the vertical axis and those that look the same when reflected across both axes. This leads to defining these types of functions as "even" or "odd," respectively. In the following activity, students match tables to the graphs and refine their definitions, so groups should keep their copies of the blackline master from this activity to use in the next activity.

Monitor for different ways groups choose different categories, but especially for categories that distinguish between graphs of even functions and graphs of odd functions. As students work, encourage them to refine their descriptions of the graphs using more precise language and mathematical terms (MP6).

If students focus too much on identifying specific points on the graph to use to make their categories, consider saying:

• “Tell me more about why you put these cards into a category together.”

• “How could looking at the overall shapes of the graphs help you sort the cards in other ways?”

Direct students’ attention to the reference created using Collect and Display. Ask students to share their categories and how they sorted their graphs. Choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between graphs of even functions and graphs of odd functions. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

It is possible students will think of graphs of odd functions as ones where a rotation using the origin as the center of rotation results in the same graph. While it is true that this type of rotation appears the same as successive reflections of the graph across both axes, focus the conversation on thinking in terms of reflections since the function notation students will use to describe odd functions, , algebraically describes reflections.

Next, display the graphs of the even functions next to the odd functions. Tell students that functions whose graphs look the same when reflected across the -axis are called even functions. Functions whose graphs look the same when reflected across both axes are called odd functions.

The purpose of this activity is for students to deepen their understanding of even and odd functions. Using the graphs from the previous activity, students first match each graph to a table of coordinate pairs and then use both representations to identify defining characteristics as they study the structure of even functions and odd functions, using function notation (MP7). 

Monitor for students making connections between the transformations described in the previous activity (a reflection across the -axis versus successive reflections across both axes) and the coordinates in the tables to share during the whole-class discussion.

Note that students may make observations about even and odd functions that are true when the functions are continuous, but may not be true for functions that are piecewise, discrete, or have discontinuities. It is not necessary at this time to clarify these observations, but it may be helpful to note that the algebraic definition for even and odd functions is true whether the function is continuous or not. 

The goal of this discussion is for students to move from observations about specific even and odd functions to generalizing things that are true for all even functions and for all odd functions.

Direct students' attention to the reference created using Collect and Display. Ask students to share observations about even and odd functions. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. As students respond, encourage them to determine whether this is true about a specific even or odd function, but not all of them, or this is true about even or odd functions in general. It is acceptable at this time if students do not consider all cases, such as piecewise or discontinuous functions, but they should be making some categorical observations.

If time allows, assign groups to write a single sentence describing even or odd functions that summarizes one of the lists.

Sample observations:

Even functions:

• Have ends that both go up or both go down.
• Can be reflected over the -axis.
• Have a line of symmetry at .
• Don't always have to cross the -axis, but do always cross the -axis.
• Have pairs of coordinates that have the same -value and opposite -values.

Odd functions:

• Have ends that go opposite directions, meaning if the left side goes up, the right side goes down, and vice versa.
• Cross at the origin.
• Can be rotated 180 degrees around the point .
• Can be reflected across both the -axis and -axis.
• Have to cross the -axis and -axis.
• Have an odd number of zeros.
• Have pairs of coordinates that have opposite - and -values.