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Linear Functions and Equations

A linear function is a function whose graph is a straight line. The line can't be vertical, since then we wouldn't have a function, but any other sort of straight line is fine. Now, are you ready to make the word "slope" a part of your life? Okay, here we go...

The following graphs show linear functions.

Positive slope.

Negative slope.

Horizontal slope.

Meanwhile, the following graphs do not show linear functions.

This graph shows a vertical line, which isn't a function.

This graph shows two lines, rather than one straight line.

This graph shows a curve, not a straight line. This graph is totally out of line.

A linear function can be described by a linear equation. A linear equation is a degree-1 polynomial. In other words, each term in a linear equation is either a constant or the product of a constant and a single variable. By the way, if you know any good-looking variables we can hook up with one of these single variables, let us know. We love playing matchmaker.

The following are linear equations:

  • x = -2
  • x + 3y = 7
  • 2x – 5y + 8 = 0

Meanwhile, the following are not linear equations:

  • xy + 7 = x + y is not a linear equation because the term xy has degree 2.
  • x + 3y2 = 6 is not a linear equation because the term 3y2 has degree 2.

While all linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value). It must also pass a polygraph test, complete an obstacle course, and provide at least three references. The qualifications are stringent.

Any equation of the form

y = (constant)

will give us a linear function.

Any equation of the form

x = (constant)

is a linear equation but does not describe a function. Remembering the absolute nonsense words "yunction" and "xquation" should help you keep things straight. Saying them out loud on the subway should help free up a seat.

Since a linear equation is just a particular kind of relation, we already know how to graph linear equations. We find some dots, then connect them. If Pee Wee can do it, so can we.

Sample Problem

Graph the linear equation y = 2x + 1.

We make a table of values, starting at x = 0 and working our way out from there along the number line:

When we graph these, we get

If we connect the dots, we get the following line:

Between any two points, there's only one way to draw a straight line. Try it yourself: draw two points, and connect them with a straight line. Can't get too creative with it, can you? No bending the paper, by the way. You don't even want to open that door.

What this rule means is that we should be able to graph any linear equation by figuring out two points and drawing the line between them. In practice, it's a good idea to graph at least three points. If we graph three points of a linear equation and they don't all lie on the same line, we know we did something wrong. As much as that might rattle our delicate egos, at least we can go back and fix what we fouled up. It's better than remaining blissfully ignorant, no matter what that old poet Thomas Gray might have said.

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