What you should learn

To interpret graphs in real-world settings

To sketch graphs for given functions

NCTM Curriculm Standards 2, 3, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Function

Vertical Axis

Horizontal Axis

Ordered Pairs

Origin

Independent Variable

Dependent Variable

Relation

Domain

Range

Introduction: The dollar value of a car begins to decrease immediately after it is sold. This is called depreciation. Mathematically, depreciation can be defined by the following open sentence.

The table below shows how a typical $15,000 car depreciates over a period of five years. From the table, you can see that as the age of the car increases, the value decreases. Why do you think this happens?

This information can also be presented in a graph. The graph below shows the relationship between the current value of the car and the age of the car.

The value of the car is said to be a function of the age of the car. A function is a relationship between input and output. In a function, the output depends on the input.

There are two number lines on the graph. The line that represents the current value of the car is the vertical axis. The line that represents the age of the car is the horizontal axis.

Notice that there are six points shown on the graph. Each point represents the dollar value of the car (in thousands) at the end of each year. For example, after 1 year, the value is $11,000. You can express this relationship as (1, 11). What do points A and B represent?

A (2, 8) means that at the end of year 2 the value of the car is about $8000.

B (3, 6) means that at the end of year 3 the value of the car is about $6000.

(1, 11), (2, 8), and (3, 6) are called ordered pairs. The order in which the pairs of numbers is written is important. Ordered pairs are used to locate points on the graph. The first number in an ordered pair corresponds to the horizontal axis, and the second number in an ordered pair corresponds to the vertical axis. The ordered pair (0, 0) corresponds to the origin.

Many real-world situation can be modeled using functions.

Exercise 1: Shim owns a farm market. The amount a customer pays for sweet corn depends on the number of ears that are purchased. Shim selss a dozen ears of corn for $3.00.

a. Make a table showing the price of various purchases of sweet corn.

b. Write four ordered pairs that represent the number of ears of corn and the price of the corn.

c. Describe a set of axes that could be used to graph the number of ears of corn and the price of the corn.

d. Draw a graph that shows the relationship between the number of ears of corn and the price.

e. As you read the graph from left to right, describe the trend you see. Explain.

 

In Exercise 1, the price a customer pays depends on the number of ears purchased. Therefore, the number of ears purchased is called the independent variable or quantity, and the price is called the dependent variable or quantity. Usually the independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.

You can use a graph without scales on either axis to show the general shape of the graph that models a situation.

Exercise 2: For a certain time, Lucinda jogs up hill at a steady speed. Then she runs down the hill and picks up her speed.

a. What happens to her speed when Lucinda jogs at a steady pace? When Lucinda runs down the hill?

The speed remains the same when her jogging pace is steady. The speed increases when she runs downhill.

b. Identify the independent and dependent quantities.

Time is the independent quantity and speed is the dependent quantity.

c. Draw a graph representing her running uphill and then downhil.

 

A relation is a set of ordered pairs. The set of first numbers of the ordered pairs is the domain of the relation. The domain contains values of the independent variable. The set of second numbers of the ordered pairs is the range of the relation. The range contains values of the dependent variable.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 9 - 15 odd, 16, 17, 19 - 23

Alternative Homework: Enriched: 8 - 14 even, 16 - 23

Extra Practice: Students book page 758 Lesson 1-9

Extra Practice Worksheet: Click Here.

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