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.