In this lesson, you’ll learn what slope is, and see some examples of positive slope. You’ll also learn how to measure a line’s slope on a graph. Then, you can test your new knowledge with a brief quiz.
What Is Slope?
What do you think of when you hear the word ‘slope’? You might think of a familiar hill that you used to hike up or roll down as a child. You might even think of a mountainside rising from ground level, such as this one. These Klondike hikers definitely know the meaning of slope! We often use the word ‘slope’ to describe how quickly a surface either rises or falls as it moves along in the horizontal direction. The more quickly it changes, the steeper the slope.
In math, the meaning isn’t much different. Slope is simply how much the graph of a line changes in the vertical direction over a change in the horizontal direction. Because of this, the slope is sometimes referred to as the rate of change.
Slopes can be positive or negative. A positive slope moves upward on a graph from left to right. A negative slope moves downward on a graph from left to right. It’s important to note that we work from left to right when dealing with slope.
Now that you know what slope is, how exactly do we calculate the slope of a line?
Let’s put that Klondike mountainside on a graph so we can measure its slope.
To calculate the slope of the line, we need the coordinates of two points that lie on that line. This graph is arranged so that the mountainside starts at (0, 0) and ends at (13, 9).
To find the rate of change, we need to calculate the rise, or the change along the y-axis, versus the run, which is the change along the x-axis. In other words, we need to find the difference between the y-coordinates and divide it by the difference between the x-coordinates. The formula we’ll use is:
S = (y2 – y1) / (x2 – x1)
Here, x1 and y1 are the x and y coordinates from the first point, and x2 and y2 are the x and y coordinates from the second point.
Since we’re going left to right, we’ll use (0, 0) as the first point and (13, 9) as the second point. So, y2 is 9, y1 is 0, x2 is 13, and x1 is 0. We can plug those values into the formula to get:
S = (9 – 0) / (13 – 0)
S = 9/13
The mountainside has a positive slope of 9/13, which means that for every 13 units the hikers move to the right, they’ll go up the hill by 9 units.
Now, what would happen if the hikers got tired and walked back down the mountainside? The mountain wouldn’t change, so neither should our slope. Let’s see what happens when we reverse the order of the points in our formula:
S = (0 – 9) / (0 – 13)
S = -9/-13, or 9/13
The slope is still 9/13. What this shows is that the slope of a line is independent of the direction in which we travel. No matter which points we use from a line, or in which order we use them, we still get the same slope.
It’s important to notice that if a line has a positive slope, the changes in x and y will always have the same sign. For example, as the hikers moved along the x-axis in a positive direction, they also moved along the y-axis in a positive direction. When they turned around and traveled in the negative x-direction, they also descended in the negative y-direction.
Examples of Positive Slope
Let’s take a look at more examples. This line on the graph also has a positive slope since it moves upward from left to right.
Just at first glance, it seems quite steep. Let’s see how steep it really is by calculating its slope.
We can use any two points on the line to determine its slope. We’ll use (0, 0) as our first point because that will make calculations easy. We’ll use (1, 10) as our second point.
That means y2 is 10, y1 is 0, x2 is 1, and x1 is 0. Let’s plug those values into our slope formula:
S = (10 – 0) / (1 – 0)
S = 10 / 1, or 10
This line has a slope of 10. It really is steep! And it definitely has a positive slope.
Let’s take a look at another line.
This one clearly has a smaller slope than the previous one. Let’s find out exactly how small.
We’ll make (0, 2) our first point and (10, 3) our second point.
S = (3 – 2) / (10 – 0)
S = 1/10
The slope of this line is definitely a small slope but still positive.
Although this lesson focuses on positive slope, it will be handy to briefly look at negative slope. At a quick glance, we see this line has a negative slope because it moves downward as it moves from left to right.
We can still use the same formula to determine this line’s slope. We’ll use (0, 9) as our first point and (9, 0) as our second point.
S = (0 – 9) / (9 – 0)
S = -9 / 9
S = -1
This line has a negative slope because a positive change in the x-coordinate results in a negative change in the y-coordinate. If a line has a negative slope, the changes in x and y will always have different signs.
Slope is how much the graph of a line changes in the vertical direction over a change in the horizontal direction. If a line has a positive slope, the changes in x and y will always have the same signs, and it will move upward from left to right. If a line has a negative slope, the changes in x and y will always have different signs, and the line will move downward from left to right.
The formula to calculate slope is:
S = (y2 – y1) / (x2 – x1)
We can use any two points on a line to calculate its slope.