Interpreting graphs in context

• Interpret key features of graphs in terms of the quantities represented (F-IF.B.4$^\star$). • Sketch graphs showing key features (F-IF.B.4$^\star$). • Relate the domain of a function to its graph (F-IF.B.5$^\star$). • Relate the domain to the quantitative relationship it describes (F-IF.B.5$^\star$).

In this section, students read and interpret graphs of functions. These include graphs of functions that arise from data, including functions that arise from using trend lines to join data points. Students also have an opportunity to apply their understanding of domain and range.

The tasks in this section could be approached as a jigsaw activity. In small groups, students become experts on one particular task, then work in new groups which include at least one expert for each task.

1 Oakland Coliseum

WHAT: Students are given information about the seating capacity and cost of tickets at the Oakland Coliseum. The revenue for the Raiders’ organization is a function of the number of people in attendance. Students are asked to find the domain and range of this function.

WHY: This task presents a real-world context for domain and range, and an opportunity for students to apply the notion of domain as all possible input values and the range as all possible output values. Among other considerations, negative and non-integer values are not included in both the domain and range.

2 Warming and Cooling

WHAT: Students are presented with a graph showing temperature change as a function of time. They are asked to use the graph to estimate values of the function, determine when the temperature is decreasing, and determine when the temperature is below a given value.

WHY: In section 2, students began to go from statements in function notation to locating the corresponding feature on graphs; here students build on those experiences to begin going in the opposite direction: examining features of a graph to in order make statements about the function it represents (MP4).

3 Influenza epidemic

WHAT: Students are presented with a graph that represents the number of people infected by influenza as it spreads across a city. Students interpret the graph, estimate values of the function represented, and explain their significance. The last part of the task tells the students that the function in the graph is $f(w) = 6w(1.3)^{−w}$, and asks students to use the graph to estimate a solution for $6w(1.3)^{−w}\geq 6$ and explain its significance for the epidemic.

WHY: The principal purpose of this task is to probe students’ ability to correlate symbolic statements about a function using function notation with a graph of the function, and to interpret their answers in terms of the quantities between which the function describes a relationship (MP7)

4 How is the Weather?

WHAT: Students are given three graphs and three statements, where each statement is represented by one of the graphs, and asked to match graphs and statements.

WHY: Students must attend to key features of the graphs in order to match them with the given statements. At first glance, the key features of the first and third graphs look similar; students must notice that the scales of the vertical axes are different (MP6).

5 Telling a Story With Graphs

WHAT: Three graphs are presented which give temperature, solar radiation, and accumulated precipitation for February 2012. Students are asked to describe the information given by each graph, then to give a story that combines information from the three graphs.

WHY: As in How is the Weather?, students interpret key features of graphs and interpret them in context. In contrast to How is the Weather?, the domains of the three functions overlap giving students the opportunity to analyze and interpret information to draw conclusions about those days (MP4). Because the graphs have different scales on their horizontal axes, students need to attend to the labels on those axes when combining information (MP6). Similarly, students need to note that the third graph shows accumulated precipitation rather than amount of precipitation.