The mathematical definition of slope is very similar to our everyday one. In math, slope is used to describe the steepness and direction of lines.  In this section, we will begin exploring the concept of slope using a geoboard and then we will learn how to measure the slopes of lines on a rectangular coordinate system.

Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. We’ll start by stretching a rubber band between two pegs to make a line as shown in the image below.

Does it look like a line?

Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in the image below. We carefully make a [latex]90^ \circ[/latex] angle around the third peg, so that one side is vertical and the other is horizontal.

To find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown below.

To help remember the terms, it may help to think of the images shown below.

On our geoboard, the rise is [latex]2[/latex] units because the rubber band goes up [latex]2[/latex] spaces on the vertical leg. See the image below.

What is the run? (Be sure to count the spaces between the pegs rather than the pegs themselves!) The rubber band goes across [latex]3[/latex] spaces on the horizontal leg, so the run is [latex]3[/latex] units.

The slope of a line is the ratio of the rise to the run. So the slope of our line is [latex]{\Large\frac{2}{3}}[/latex]. In mathematics, the slope is always represented by the letter [latex]m[/latex].

What is the slope of the line on the geoboard in the image above?

[latex]m={\Large\frac{\text{rise}}{\text{run}}}[/latex]
[latex]m={\Large\frac{2}{3}}[/latex]

The line has slope [latex]{\Large\frac{2}{3}}[/latex]

When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.

If we start by going up when we connect the second peg, the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.

Since the slope formula is rise over run, it may be easier to always count out the rise first and then the run.

Finding the Slope of a Line from its Graph

By just looking at the graph of a line, you can learn some things about its slope, especially relative to other lines graphed on the same coordinate plane. Consider the graphs of the three lines shown below:

First, let’s look at lines A and B. If you imagined these lines to be hills, you would say that line B is steeper than line A. Line B has a greater slope than line A.

Next, notice that lines A and B slant up as you move from left to right. We say these two lines have a positive slope. Line C slants down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the rise and the run. The vertical change between two points is called the rise, and the horizontal change is called the run. The slope equals the rise divided by the run: [latex]\displaystyle \text{Slope }=\frac{\text{rise}}{\text{run}}[/latex].

You can determine the slope of a line from its graph by looking at the rise and run.  (Notice the similarity between the image above and the Geoboard examples we looked at earlier.) One characteristic of a line is that its slope is constant all the way along it. So, you can choose any 2 points along the graph of the line to figure out the slope. Let’s look at an example.

This line will have a slope of [latex]\displaystyle \frac{1}{2}[/latex] no matter which two points you pick on the line. Try measuring the slope from the origin, [latex](0,0)[/latex], to the point [latex](6,3)[/latex]. You will find that the [latex]\text{rise}=3[/latex] and the [latex]\text{run}=6[/latex]. The slope is [latex]\displaystyle \frac{\text{rise}}{\text{run}}=\frac{3}{6}=\frac{1}{2}[/latex]. It is the same!

Let’s look at another example.

When you look at the two lines, you can see that the blue line is steeper than the red line. It makes sense the value of the slope of the blue line, [latex]4[/latex], is greater than the value of the slope of the red line, [latex]\displaystyle \frac{1}{4}[/latex]. The greater the slope, the steeper the line.

Finding the Slope of a Line From a Graph

Distinguish between graphs of lines with negative and positive slopes

Direction is important when it comes to determining slope. It’s important to pay attention to whether you are moving up, down, left, or right; that is, if you are moving in a positive or negative direction. If you go up to get to your second point, the rise is positive. If you go down to get to your second point, the rise is negative. If you go right to get to your second point, the run is positive. If you go left to get to your second point, the run is negative.

In the following two examples, you will see a slope that is positive and one that is negative.

The next example shows a line with a negative slope.

In all of the previous examples of finding the slope of a line, we were given two points.  Now we will look at some examples where we are not automatically given two points on the line.

To find the slope, we must count out the rise and the run. But where do we start?

We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.

example

Find the slope of the line shown:

Solution
Locate two points on the graph, choosing points whose coordinates are integers. We will use [latex]\left(0,-3\right)[/latex] and [latex]\left(5,1\right)[/latex].

Starting with the point on the left, [latex]\left(0,-3\right)[/latex], sketch a right triangle, going from the first point to the second point, [latex]\left(5,1\right)[/latex].

Count the rise on the vertical leg of the triangle.	The rise is [latex]4[/latex] units.
Count the run on the horizontal leg.	The run is [latex]5[/latex] units.
Use the slope formula.	[latex]m={\Large\frac{\text{rise}}{\text{run}}}[/latex]
Substitute the values of the rise and run.	[latex]m={\Large\frac{4}{5}}[/latex]
The slope of the line is [latex]{\Large\frac{4}{5}}[/latex] .

Notice that the slope is positive since the line slants upward from left to right.

example

Find the slope of the line shown:

Show Solution

Solution
Locate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use [latex](0, 5)[/latex] and [latex](3, 3)[/latex]. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.

Count the rise – it is negative.	The rise is [latex]−2[/latex].
Count the run.	The run is [latex]3[/latex].
Use the slope formula.	[latex]m=\Large\frac{\text{rise}}{\text{run}}[/latex]
Substitute the values of the rise and run.	[latex]m={\Large\frac{-2}{3}}[/latex]
Simplify.	[latex]m=-{\Large\frac{2}{3}}[/latex]
The slope of the line is [latex]-{\Large\frac{2}{3}}[/latex].

Notice that the slope is negative since the line slants downward from left to right.

What if we had chosen different points? Let’s find the slope of the line again, this time using different points. We will use the points [latex]\left(-3,7\right)[/latex] and [latex]\left(6,1\right)[/latex].

Starting at [latex]\left(-3,7\right)[/latex], sketch a right triangle to [latex]\left(6,1\right)[/latex].

It does not matter which points you use—the slope of the line is always the same. The slope of a line is constant!

The lines in the previous examples had [latex]y[/latex] -intercepts with integer values, so it was convenient to use the y-intercept as one of the points we used to find the slope. In the next example, the [latex]y[/latex] -intercept is a fraction. The calculations are easier if we use two points with integer coordinates.

example

Find the slope of the line shown:

Show Solution

Solution

Locate two points on the graph whose coordinates are integers.	[latex]\left(2,3\right)[/latex] and [latex]\left(7,6\right)[/latex]
Which point is on the left?	[latex]\left(2,3\right)[/latex]
Starting at [latex]\left(2,3\right)[/latex] , sketch a right angle to [latex]\left(7,6\right)[/latex] as shown below.

Count the rise.	The rise is [latex]3[/latex].
Count the run.	The run is [latex]5[/latex].
Use the slope formula.	[latex]m=\Large\frac{\text{rise}}{\text{run}}[/latex]
Substitute the values of the rise and run.	[latex]m={\Large\frac{3}{5}}[/latex]
The slope of the line is [latex]{\Large\frac{3}{5}}[/latex].

In the following video we show another example of how to find the slope of a line given a graph. This graph has a positive slope.

In the following video we show another example of how to find the slope of a line given a graph. This graph has a negative slope.

Finding the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

• horizontal line [latex]y=b[/latex]; all the [latex]y[/latex] -coordinates are the same.
• vertical line [latex]x=a[/latex]; all the [latex]x[/latex] -coordinates are the same.

So how do we find the slope of the horizontal line [latex]y=4?[/latex] One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens. We’ll use the two points [latex]\left(0,4\right)[/latex] and [latex]\left(3,4\right)[/latex] to count the rise and run.

The slope of the horizontal line [latex]y=4[/latex] is [latex]0[/latex].

All horizontal lines have slope [latex]0[/latex] . When the [latex]y[/latex] -coordinates are the same, the rise is [latex]0[/latex] .

Now we’ll consider a vertical line, such as the line [latex]x=3[/latex], shown below. We’ll use the two points [latex]\left(3,0\right)[/latex] and [latex]\left(3,2\right)[/latex] to count the rise and run.

But we can’t divide by [latex]0[/latex]. Division by [latex]0[/latex] is undefined. So we say that the slope of the vertical line [latex]x=3[/latex] is undefined. The slope of all vertical lines is undefined, because the run is [latex]0[/latex].

The following example shows you how to determine the slope of horizontal and vertical lines that are plotted on the coordinate axes.