# Using Points to Determine the Shape of a Graph

## Objective

SWBAT create a table of values that will help them determine the shape of the graph of a real-world situation.

#### Big Idea

In order to get a broad picture of functions and their graphs, we go a bit beyond just the linear and exponential functions that are the foci of this unit.

## Opener: How to Use a Table to Think About the Bus Problem

15 minutes

Today's Opener jumps off from the assumption that students buy what they saw in the Everyday Situations handout and we've accurately matched the four algebraic rules to the descriptions of four situations.  From there, we can really think about how an algebraic rule can help reveal what a graph should look like.

As I've described in previous lessons, I really like the arc of the "Everyday Situations" assignment.

Today's Opener

With all of that in mind, the opener references one of the situations on the "Everyday Situations" assignment:

A group of people rent a bus for a day.  The total cost of the bus is shared equally among all the passengers.  x = the number of passengers; y = the cost for each passenger in dollars

Students had trouble coming up with what the graph of this situation should look like.  On today's opener, I add what we learned from the algebraic rule that matches this situation: that the cost of the bus rental is \$300.  Then, I instruct students to "make a table of values showing how much everyone would have to pay if there were 1, 2, 3, 4, 5, 6, 10, 15 or 30 passengers.

Making the table is an accessible task.  I can count on almost all of my students jumping right into this one.  I circulate to make sure they've got it, and then as they work, I prepare some axes on the front board.  After a few minutes, I elicit help from the class to fill in the table, and then I say that we're going to use these points to learn what the graph of this situation should look like. As soon as we plot the points, the shape of this function becomes clear.

Now we continue with the "Car Value" situation:

My car loses about half of its value each year.  x = the time that has elapsed in years; y = the value of my car in dollars.

As an example of exponential decay, this function actually gets at the focus of this unit.  We follow the same process, making a table of values and seeing the shape of the curve.

At a few points during this mini-lesson, I summarize for students by saying, "There are two things I want you to think about as you consider each situation.  First, as x increases, does y increase or decrease?  Second, should the graph be a straight line or a curve?"