## Lesson: Functions and Graphing Developing the Concept

Now that students have reviewed how to graph points, it's time to have them graph some linear relationships and interpret a graph in terms of a problem situation.

Materials: graph paper and pencil

Preparation: none

Prerequisite Skills and Background: Students should be able to graph points on a coordinate plane and determine if points are close to a graph.

• Ask: Does anyone know how long it takes world-class male runners to run the Boston Marathon? (a little over two hours)
• Say: That's right; they run more than 26 miles in a little over two hours. Since marathoners run at a fairly steady pace, this means they run at a rate of about 13 miles an hour. We can get a graph of the approximate distance a world-class runner will have run in a marathon at any given time by graphing the equation: d = 13t, where d equals the distance in miles and t equals the time in hours.

Write the equationd = 13t on the board.

• Ask: How would I find out the distance a runner would have traveled if he had been running for a half-hour? (Substitute 0.5 or into the equation for t.) Everyone do that at your desk and tell me the approximate distance he would have run. (6.5 miles)
• Say: Now find the approximate distance a runner would have run in 0.7 hour, 1 hour, and 1.6 hours. (9.1 miles, 13 miles, and 20.8 miles, respectively)

Make a table and put the entries for 0.5 hours, 0.7 hours, 1 hour, and 1.6 hours in it and their corresponding distances.

• Say: Now let's graph these points.
Create a graph similar to the one below.
• Say: I'd like you to create a graph similar to this at your desks.
Walk around the room helping students who are having difficulty.
• Ask: What do you notice about the four points that you graphed?
They seem to lie in a straight line.
• Say: They do lie in a straight line. Equations like d = 13t are called linear equations because their graphs are straight lines. Let's connect all the points starting at (0, 0) with a straight line.
• Ask: Based on your graph, approximately how far would a world-class male marathoner run in 1.2 hours? (15.5 miles) How did you get your answer?
(I looked along the x-axis to find the time, 1.2 hours. I then looked directly up to the graph. Then I looked horizontally across to the y-axis to find the distance, about 15.5 miles.) Remind students that their answers will be close, but not exact.
• Say: That's great. What if I told you that a world-class male marathoner could run 22 miles in either 1.4 or 1.7 hours? How could you find out which was a more accurate time?
The time of 1.7 hours is more accurate, because if we find the distance 22 miles on the y-axis and read across horizontally and then down to the time on the x-axis, it is closer to 1.7 hours than it is to 1.4 hours.
• Ask: How long does it takes for a world-class male marathoner to run about three miles? (a little less than 15 minutes) How did you figure that out?
(I looked at 3 miles on the y-axis and read across to the graph and then looked down to the time in hours on the x-axis; it was a little less than 0.25 hour.)

Wrap-Up and Assessment Hints
If a linear equation such as y = 3x + 2 is represented by either a graph or the equation, you can assess your students' understanding by asking questions like the following.

If x increases 1 unit, what happens to y? (It increases 3 units.)
If x = 0, what is the value of y? (2)
If x increases 4 units, what happens to y? (It increases 12 units.)
If y decreases 6 units, what happens to x? (It decreases 2 units.)
For what values of x is y negative? (x < -)
For what values of x is y > 8? (x > 2)