The purpose of this activity is for students to connect different function representations and learn the conventions used to label a graph of a function. Students first match function contexts and equations to graphs. They next label the axes and calculate input-output pairs for each function. The focus of the discussion should be on what quantities students used to label the axes and recognizing the placement of the independent or dependent variables on the axes.

Monitor for students who recognize that there is one graph that is not linear and match that graph with the equation that is not linear.

The purpose of this discussion is for students to understand the conventions of constructing a graph of a function and where input and outputs are found on a graph. Select previously identified students to share how they figured out matched the nonlinear graph.

Ask students:

• “Where are the independent variables labeled on the graphs?” (the horizontal axis)
• “Where are the dependent variables labeled on the graphs?” (the vertical axis)

Tell students that by convention, the independent variable is on the horizontal axis, and the dependent variable is on the vertical axis. This means that when we write coordinate pairs, they are in the form of (input, output). For some functions, like the one with quarters and dimes, we can choose which is the independent variable and which is the dependent variable, which means the graph could be constructed either way based on our decision.

Conclude the discussion by asking students to share their explanations for the point for Figure C. (There is no such thing as 0.6 of a quarter.) Remind students that sometimes we have to restrict inputs to only those that make sense. Since it’s not possible to have 49.6 quarters, an input of 1 dime does not make sense. Similarly, 2, 3, or 4 dimes result in numbers of quarters that do not make sense. 0 dimes or 5 dimes, however, do produce outputs that make sense. Sometimes it is easier to sketch a graph of the line even when graphing discrete points would be more accurate for the context. Keeping the context of a function in mind is important when making sense of the input-output pairs associated with the function.

The purpose of this activity is for students to interpret coordinates on graphs of functions and nonfunctions as well as understand that context does not dictate the independent and dependent variables.

In the first problem, time is a function of distance since each input of meters ran has one and only one output of seconds past. The graph and table help determine how long it takes for Kiran to run a specific distance. In the second problem, time is not a function of Priya’s distance from the starting line since this graph includes her distance from the starting line as she returns to the starting line. This results in a graph in which each input does not give exactly one output.

The purpose of this discussion is for students to understand that independent and dependent variables are not determined by the context (and specifically that time is not always a function of distance). Select students to share their strategies for calculating the answers for the first set of problems. For each problem, ask students whether the graph or table was more useful. Further the discussion by asking:

• “When are tables useful for answering questions?” (We can get exact values from the numbers in the table, while with the graph, we can only approximate.)
• “When are graphs more useful for answering questions?” (A line graph shows more input-output pairs than a table can list easily.)
• “Why does it make sense to have time be a function of distance in this problem?” (The farther Kiran runs, the longer it will take, so it makes sense to represent time as a function of distance.)
• “Does time always have to be a function of distance?” (No, this graph could be made the other way with time on the horizontal axis and distance on the vertical axis, and then it would show distance as a function of time, which makes sense since the longer Kiran runs, the farther he will travel.)

For the second graph, ask students to indicate if they think it represents a function or not. If there are students who say yes and no, invite students from each side to share their reasoning and try to persuade the rest of the class to their side. If all students are not persuaded that the graph is not a function, remind them that functions can only have one output for each input, yet the answer to the problem statement “Estimate when she was 100 meters from her starting point” has two possible responses, 18 seconds or 54 seconds. Since that question has two responses, the graph cannot represent function.