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Functions - Linear Functions and Equations

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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.

  • Intercepts

    The intercepts of a linear equation are the places where the axes catch the pass thrown by the linear equation. This is our effort to make linear equations seem remotely athletic. In reality, they have about as much physical ability as Tim Tebow. Oh, snap.

    (Shmoop is strongly partisan about football, in case you couldn't tell.)

    In non-sports-analogy terms, the intercepts are the spots at which the axes and the graph of the linear equation overlap one another. The x-intercept is the place where the graph hits the x-axis, and the y-intercept is the place where the graph hits the y-axis. It would be awfully confusing if it were the other way around.

    A linear equation may have one or two intercepts. Sometimes either the x-intercept or the y-intercept doesn't exist, or so intercept atheists would have you believe.

    Knowing both intercepts for a linear equation is enough information to draw the graph, provided the intercepts aren't 0. If they are 0, then our graph could be drawn any which way.

    Sample Problems

    Draw the graph of the linear equation with x-intercept 3 and y-intercept 4.

    First we draw points at the intercepts:

    Then we connect the dots:

    If the graph goes through the origin (0, 0), then both of the intercepts are 0 and we don't have enough information to draw the graph. We even tried calling 411, but they acted as if they had no idea what we were talking about.

  • Slope

    The slope of a linear equation is a number that tells how steeply the line on our graph is climbing up or down. If we pretend the line is a mountain, it's like we're talking about the slope of a mountain. If it helps you, draw a snowcap at the top. Some mountain climbers. A ski lift. Nothing too elaborate though.

    We move from left to right on the x-axis, the same way that we read. If the line gets higher as we move right, then we're climbing the mountain, so the line has a positive slope

    If the line gets lower as we move right, then we're descending the mountain, so the line has a negative slope.

    If we stay at the same height, then the slope is zero because we're not going up and we're not going down. Pretty boring mountain, if you ask us.

    Now let's find some actual numbers for slopes. Thinking of the mountains, a slope is a ratio that describes how quickly our height changes as we move over to the right. Not our actual physical height, mind you. We won't be getting shorter or taller throughout the course of these examples, even if you do feel by the end of it that you've grown.

    Sample Problem

    Julie is climbing a mountain. For every 10 feet Julie travels (measured along the ground), she goes 20 feet higher. What is the slope of the mountain?

    The slope of the mountain is .

    For every foot Julie travels (measured along the ground), she gets 2 feet higher off the ground. She'd be even higher off the ground if she'd worn heels, but we suppose those would have been an odd choice for mountain climbing.

    Check this out. Look at the graph of the line y = x:

    The slope of the line y = x is 1. If we move over to the right by 1 on the x-axis, we also move up by one on the y-axis:

    Sample Problem

    Find the slope of the line pictured below. If we haven't heard from you in three hours, we'll send the park ranger after you.

    Let's look at what happens between a couple points of the graph:

    On this line, or mountain, we move up 2 for every 3 we move over. Except for that one time we moved up 2, encountered a mountain lion, and ran back down 7.

    Our slope is .

    One way to think about slope is 

    To use this formula to find the slope of a line, we first fix two points on the graph whose coordinates we can easily figure out.

    The rise is the amount y changes between those two points, and this number may be positive or negative. Remember, you can be going up or down the mountain. The run is the amount x changes between those two points. We usually think of moving from the point on the left to the point on the right, meaning that x is increasing and the "run'' is always positive. You might climb up or down, but you would never run backwards, right? Aside from when you were backing away from that mountain lion, we mean.

    Then the slope of this line is:

    Be careful: It's all very well and good to memorize the formula , but in order to use it correctly, you need to know what "rise'' and "run'' really mean. It doesn't refer to your underwear rising up on you or your stockings having a run in them, although either would be a wonderfully memorable image. In addition to the formula, it might be helpful to have a picture like the one below in your head:

    Sample Problem

    Find the slope of the line shown below.

    Let's find a couple of points whose coordinates are nice and easy to work with and see what the rise and run are between those two points. Use the undergarment visual if you'd like. It won't help you with this problem, but no one's stopping you.

    The slope is:

    Sample Problem

    Find the slope of the line shown below.

    If we try to apply the formula   to a vertical line, we'll be in trouble. Since the "run'' between any two points on a vertical line is 0, and we can't divide by 0, the slope of a vertical line is undefined. So, the slope of the line x = 1 is undefined. 

     Makes sense, since it would take some powerful thighs to run directly up a vertical mountain. If you attempted to do so, you'd soon be undefined as well.

    Sample Problem

    Find the slope of the line that passes through the points (1, 3) and (2, 7).

    We can find the slope of a line if given any two points on the line. We know part of the line will look like this:

    To get from the point (1, 3) to the point (2, 7), we need to move right 1 and up 4:

    That means the slope of the line is . Yodelay-hee-hoo!

    Sample Problem

    Find the slope of the line that goes through (-3, 1) and (2, -2).

    Part of the line looks like this:

    The distance we travel to get from one value of x to the other is 3 + 2 = 5, since first we have to travel from x = -3 to x = 0 and then from x = 0 to x = 2. We have a layover at the y-axis, where we can grab a quick bite of vastly overpriced fast food while we wait for our connecting line.

    To get from one value of y to the other, first we travel from y = 1 to y = 0 and then from y = 0 to y = -2, for a total rise of -3. Once again, we couldn't get a direct flight. Ah, well. It'll give us more time to read this book we've been working on.

    Thus the slope of this line is .

    Be careful: It's common to make mistakes calculating the rise and run when there are negative coordinates involved. To avoid mistakes, we recommend drawing a picture to help with the calculations. If art isn't your thing, find a mountain or book a flight so you can live out one of our previous examples. More expensive and time-consuming to get the point across that way, but it'll certainly drive the idea home.

    Well, now we can read off the slope of a line from a graph or from any two points on the line. We're feeling good about ourselves. How about graphing a line if given a single point and a slope?

    Sample Problem

    Graph the line that goes through (0, 0) and has a slope of 2.

    Let's start by drawing the point we're given:

    We're told the line has a slope of 2, which means as x moves over 1, y goes up 2:

    We now have two points, which is enough to draw a line:

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