Intent Students will experiment with “families” of functions and use graphing calculators to prepare presentations. The activity will strengthen their understanding of the connections among equations, tables, and graphs.

Mathematics This activity is an informal introduction to transformations of functions. Students will begin to gain some experience with the graphical representations of function families like y = mx + b, y = a(x – b)² + c and y=ax−b+c size 12{y=a sqrt {x-b} +c} {}. They will begin to build a set of notes that includes the various representations of each of these function families.

Progression Students will work on this open-ended exploration in groups and then add to their own discoveries those of other students in the class.

Approximate Time 95 minutes for activity, presentations, and discussion

Classroom Organization Groups and whole class

Materials Poster-size grid paper

Doing the Activity Introduce the activity and explain that, as part of groups’ general exploration, they must explore the graphs of each of the following three functions and study variations on at least two of the three. y = xy = x² y=x size 12{y= sqrt {x} } {}Discuss what you mean by “variations.” For instance, variations on the function y = x include y = 5x and y = x + 1, and variations on y=x size 12{y= sqrt {x} } {}include y=2x size 12{y=2 sqrt {x} } {}, y=5x size 12{y= sqrt {5x} } {}, and y=x+3 size 12{y= sqrt {x+3} } {}. Also discuss the write-up and presentation aspects of the activity. As groups work, assign specific functions or families for groups to prepare reports and posters for, which will help illustrate the general shapes of graphs in the various categories. Groups should prepare these reports and posters as they go, rather than waiting until they have completed the entire exploration. Each poster should show an equation, its graph and the viewing rectangle used, and an In-Out table. You may want to urge students to use function notation as well as “y =” notation.

Discussing and Debriefing the Activity As groups present, hang the posters to create a display of various functions and their graphs around the room. You might have groups present their findings about their function families and then have the class enter certain equations in their calculators to view the families up close. Presenters should write down what should be entered into the calculators to create a given graph type. Be aware that students may use different viewing rectangles, in which case they will get different-looking graphs for the same equation. Allow enough time after each presentation for audience members to add the sketch and other information to their notes. Each student can thereby create his or her own list of matching equations and graphs. After all the function families have been introduced, focus on ways to think about the different families. Ask, How can you organize all of these graphs in a systematic way? Have the class rearrange the posters, grouping similar graphs together. Let students decide what “similar” means, as well as what other criteria to use for this organizing process. It may be helpful to consider x-intercepts, y-intercepts, and shape.

Key Question How can you organize all of these graphs in a systematic way?

Supplemental Activity Family of Curves (reinforcement or extension) is a follow-up to this activity.Give each group the equation of a basic curve, such as y = x², and ask them to look at some simple changes that could be made to the equation, such as y = x² + 2 or y = 3x². Groups should then explore how graphs vary among members of that family of curves and make a poster showing their results. They can use graphing calculators or computers to assist with their work.