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There are two special cases when it comes to slopes on the xy plane: horizontal and vertical lines. Without any more information, these examples can be pretty confusing. But with a little instruction, they end up being some of the easiest lines to graph!

Slope Basics & Review

In this lesson, you'll take a look at some oddball slopes. But first, let's review the basics. Slope is the amount of vertical change per unit of horizontal change. In other words, as a line moves one unit to the right, how many units does it go up or down?

You learned in another lesson that if a line goes up from left to right, its slope is positive, and if it goes down from left to right, its slope is negative. You also know how to identify whether the slope is positive or negative when the line is written in y = mx + b form. M represents the slope of the line, so if m is positive, the slope is positive, and the line will be slanting upwards. If m is negative, the slope is negative, and the line will be slanting downwards.

But what if you get one of these?

In this lesson, you'll learn how to deal with both of those cases. They might look tricky when you first start out, but they're not actually that bad once you get to know them - in fact, they're some of the easiest slopes to handle!

Undefined Slope

First, we'll start with this one.

If the line is vertical, it means that the slope is undefined: it has no value that we can express in numbers. That's a pretty crazy concept, so let's take a look at what's going on here. You know that as a line gets steeper and steeper, the slope gets bigger and bigger if it's positive, or smaller and smaller if it's negative. In both cases, the absolute value of the slope increases as it gets steeper.

The bigger a number gets, the steeper the slope is. But the problem with numbers is that they can just keep getting bigger and bigger. There's no such thing as the biggest number. So, no matter how steep the slope is, there will always be a slope that's even steeper. You can just add or subtract 1.

To get a completely vertical slope, you'd have to have a number that was simultaneously the biggest and the smallest number in existence, but that's not possible. Neither of those numbers exist, and they certainly can't be the same number. That's why the slope is undefined. Mathematically, the slope isn't actually a real number; hence we call it undefined.

So if m is undefined, how do we write this line as an equation? If you look more closely at the graph, you'll see that the x-value is exactly the same for the entire line. So to represent this line numerically, we use the equation x = ____. The line here is the line x = -2. The x-value is the same for every value of y: it's always -2.

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Now let's look at a similar problem: what about a horizontal line? As a line gets flatter and flatter, the absolute value of the slope gets smaller and smaller. But unlike the increasing slopes, there is a limit to how much a number can decrease. There is such a thing as a number with the smallest absolute value: 0.

That's why slope of a horizontal line is 0. You can also think of this mathematically. Slope represents how many units the line goes up for every unit it moves to the right. This line doesn't go up at all, so it goes up 0 units for every 1 unit of movement to the right. That makes the slope 0/1, or 0.

If we plug this into the y = mx + b form, we get y = 0x + b. Since any number multiplied by 0 is 0, we represent this line with the equation y = ____. Another way of looking at this is to notice that the value of y is always the same.

Lesson Summary

In this lesson, you learned about the slopes of horizontal and vertical lines. Vertical lines have an undefined slope, and they're written in the form x = ___. Horizontal lines have a slope of 0, and they're written in the form y = ___.

These slopes look a little complicated when you first approach them, but once you understand the concept behind them, they're really not that hard. They rely on all the same concepts that you learned about for normal slopes; they're just the very extremes of what can possibly happen. Now try out some for yourself on the quiz questions!

Learning Outcomes

By working your way through this lesson, you should feel confident completing the following tasks:

Recall the definition for the slope of a line

State and explain the slopes of horizontal and vertical lines

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