Intent This is the first of three activities in which students look for equations that fit given graphs.

Mathematics Students are developing their skill at moving among the various representations of functions. Drawing on work in The Overland Trail, this activity asks students to find symbolic rules for three linear graphs. In the discussion, they will attend to the x- and y-intercepts of these graphs.

Progression Students will work on this activity individually and discuss their ideas as a class.

Approximate Time 5 minutes for introduction 15 minutes for activity (at home or in class) 15 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Doing the Activity Graph a may need particular attention, because its equation does not involve x. An In-Out table should be especially helpful here, as students will see that the Out values are all equal to 5.

Discussing and Debriefing the Activity During the discussion, illustrate the use of the terms x-intercept and y-intercept. Ask what the intercepts are for each graph. Graph b has an x-intercept at ( 23 size 12{ { {2} over {3} } } {}, 0) and a y-intercept at (0, –2), though students might give a decimal estimate such as (0.7, 0) for the x-intercept. The only intercept for graph c is the origin (0, 0), which is both an x-intercept and a y-intercept. Graph a has a y-intercept, at (0, 5), and no x-intercept. Let students know that an intercept is sometimes identified by a single coordinate. For instance, we might say that the y-intercept for graph b is –2. You might ask, Why don’t we need to state the x‑coordinate for a y-intercept (or vice versa)? Enrich students’ understanding by asking what the y-intercept means in a real-world context, such as those studied in The Overland Trail. They might recall that the y-intercept often gives the “starting value” for a situation. This view is particularly appropriate if the horizontal axis represents time. Finally, you might also raise the question of how many intercepts of each kind a graph might have. Students may be able to explain why a graph of a function cannot have more than one y-intercept and perhaps will be aware that it can have any number of x-intercepts.

Key Question What are the x- and y-intercepts for each graph?