Linear Functions

Student: So I have practiced guessing functions and am getting pretty good at it as long as there is one operation. The more complicated functions are harder.

Mentor: Yes, they are. The best way to understand such functions is to study one kind at a time. Let's start with functions of the form:

`Y = ____ * X + ____`

These functions are called linear functions, and are often written as:

`Y = m * X + b `

Where m represents the number multiplied to X and b represents the number added to the result.

Student: What's so important about these?

Mentor: These functions increase or decrease steadily. Look at the following function and table of points from the function:

`Y = 4 * X + 2 `

X Y
0 2
1 6
2 10
3 14
4 18

Now, answer some questions for me. What is the value of the function when X is 0?

Student: 2.

Mentor: Good. What is the change in the value of the function as X increases by 1?

Student: Well, the value of the function goes from 2 to 6 to 10. So at each step the function increases by 4.

Mentor: Now look at your answers: 2 for the starting point, when X is 0 and 4 for the increase. Do those numbers look familiar?

Student: In the original function, Y = 4 * X + 2, m = 4 and b = 2. The same numbers we got for the starting value and the increasing value. Is this a coincidence?

Mentor: No, it is not a coincidence. This always works. Try some.

Student: Here are a few:

• `Y = 10 * X - 1`
Change = 10, start = -1
• `Y = -2 * X + 3`
Change = -2, start = 3
• `Y = 5 * X + 11`
Change = 5, start = 11

Mentor: Good! But before we begin, let's get the terminology right: The change is called the slope and the starting value is called the intercept. We'll learn why these words are used later when we talk about graphs. Can you build a few tables of ordered pairs to further demonstrate these facts about your functions? You may wish to use Simple Plot to plot the ordered pairs from your table.