Lesson 4.1

What is the real-world meaning of the graph at right, which shows the relationship between the number of customers getting haircuts each week and the price charged for each haircut?

The number of customers depends on the price of the haircut.

So the price in dollars is the independent variable and the number of customers is the dependent variable.

As the price increases, the number of customers decreases linearly. As you would expect, fewer people are willing to pay a high price; a lower price attracts more customers.

The slope indicates the number of customers lost for each dollar increase.

The x-intercept represents the haircut price that is too high for anyone.

The y-intercept indicates the number of customers when haircuts are free.

Students at Central High School are complaining that the juice vending machine is frequently empty. Several student council members decide to study this problem. They record the number of cans in the machine at various times during a typical school day and make a graph.

a. Based on the graph, at what times is juice consumed most rapidly?

b. When is the machine refilled? How can you tell?

c. When is the machine empty? How can you tell?

d. What do you think the student council will recommend to solve the problem?

Although the student council members in the example are interested in solving a problem related to juice consumption, they could also use the graph to answer many other questions about Central High School:◦ When do students arrive at school?◦ What time do classes begin? ◦ When is lunch?◦ When do classes let out for the day?

Both the graph of haircut customers and the graph in the example are shown as continuous graphs.

In reality, the quantity of juice in the machine can take on only discrete values, because the number of cans must be a whole number.

The graph might more accurately be drawn with a series of short horizontal segments, as shown at right.

The price of a haircut and the number of customers can also take on only discrete values. This graph might be more accurately drawn with separate points. However, in both cases, a continuous “graph sketch” makes it easier to see the trends and patterns.

Every graph tells a story. Make a graph to go with the story in Part 1.

Then invent your own story to go with the graph in Part 2.

Sketch a graph that reflects all the information given in this story.

“It was a dark and stormy night. Before the torrents of rain came, the bucket was empty. The rain subsided at daybreak. The bucket remained untouched through the morning until Old Dog Trey arrived as thirsty as a dog. The sun shone brightly through the afternoon. Then Billy, the kid next door, arrived. He noticed two plugs in the side of the bucket. One of them was about a quarter of the way up, and the second one was near the bottom. As fast as you could blink an eye, he pulled out the plugs and ran away.”

This graph tells a story. It could be a story about a lake, a bathtub, or whatever you imagine. Spend some time with your group discussing the information contained in the graph. Write a story that conveys all of this information, including when and how the rates of change increase or decrease.