Slope of Linear Graphs - Online Tutor, Practice Problems & Exam Prep

To calculate the slope of a line, use the formula for slope, which is the rise over run, expressed as $\frac{y₂-y₁}{x₂-x₁}$. The rise indicates vertical change, while the run indicates horizontal change. A negative slope indicates a downhill trend, while a positive slope indicates an uphill trend. The steepness of the slope reflects the rate of change, with larger values indicating steeper lines.

Calculating Slope of a Straight Line

Alright. So let's continue by learning how to calculate the slope of a line. You may guys might remember this from algebra. Right? So here in the green box, I've got our formula for slope. It's going to be our rise over run, the change in y over the change in x, y2 - y1 over x2 - x1. These all mean the same thing. In this class, it's probably not going to get so algebra heavy, so we're just going to stick with rise over run for now. Cool. So let's go ahead and calculate a few examples here and see how this formula works. So let's start with this first graph A with the red line. And my first step when I'm calculating slope on a graph is I try to find two points that intersect on the graph directly on, one of these intersections on the graph. This first graph actually has quite a few of them. So you know right here, right here, right here. Right? Some of our other I'm going to go ahead and pick this point and this point. Two points, that intersect the line there, and let's go ahead and calculate the rise first and then we'll do the run. So to calculate the rise, we have to see what the change in the vertical axis is. What is the change in the y value? So if we start here, we want to see how much did the up and down change for between these two points. So we started here at 5 and it looks like the next point is down here on 4, right? So it looks like we went down 1 and when we're calculating slope down is going to be negative and up is positive. Just like when we're going left and right, left is negative and right is positive. I'll write that all down here. I'll put it here on the left hand side for you. So up is going to be positive and right is going to be positive. Down is going to be negative and left is going to be negative. Up and to the right is positive. Cool. So in our example here, like I said we went down 1, so that is going to be a slope of negative one. Excuse me, not a slope of negative one, a rise of negative one. Now let's see what the run is. So from one point to the next, the x value seems to have shifted one to the right here. So when it goes to the right it's positive right? We've got a positive one for the run. So let's go ahead and calculate the slope here. We've got slope and I'll write it rise over run. So our rise in this case was negative 1. Our run was 1, so that's gonna simplify to negative one. Our slope here is negative one. If you guys need a little refresher with fractions as well, I'm also including a fractions review, in this section too. Cool. So let's move on to, part b here, and let's calculate the slope here. I'm gonna get out of the way so we can see the example, and let's go ahead and remember, like I said, the first step, we want to find 2 points that are intersecting the graph, at one of those intersections. Right? So you can see in this case, we've got a few points here that don't exactly cross at those intersections. We want to find the two points or any two points that are crossing. It just makes it easier to calculate. So right here in the middle we've got one point and I'm going to pick this one right here on the end, and we're going to calculate the slope, between those two points. So let's first do the rise. The rise in this case. So it looks like we started at a vertical value of 2 and the next point is at a vertical value of 3. So let's see. We're going to draw our arrow here and it looks like it went up 1, from 2 to 3. So I will write 1 right here, and now let's do our run. So we started at 3 and we went to 6, so it looks like our change was 3 here, right? From 3 to 6, our run is 3. So let's go ahead and calculate the slope. Slope again, I'll write it here. Rise over run. And in this case, our rise was 1, our run was 3, and that's it. The answer is 1 third. The slope of this line is 1 third. So let's scroll down here. We've got one more graph, part c, and let's go ahead and calculate this slope. So I guess I'll come back so you don't feel so lonely. Hey guys. Alright. So let's do, part c. Again, we want to find 2 points, where it's intersecting directly, there on the graph. So notice kind of a point like that they're not easy to calculate so let's find the easy points. I'll do it in blue. We've got one right here and one right here. There's other ones, but those are the ones I'm going to use. Cool. So let's start with our rise again. In this case, we start at 4. Our next, vertical value is 6. So it looks like we're going to go up here, and it looks like we went up 2. Right? We started at 4, went to 6. So our rise was 2. Let's do the same thing with our run. In this case it looks like we started at 3 and we got to 4. So it looks like our run is going to be 1 in this case, and let's calculate the slope. So our slope, again, rise over run. Right? And our rise was 2. Our run was 1. 2 over 1, that's just 2. So our slope in this case is 2. So let's go ahead and compare, just let's look at these lines and see the difference in the slope and what the line looks like. So in part A, we've got a negative slope, right? Our slope was negative one and notice how this line looks compared to the other lines, right? It looks like when we go from left to right, it looks like this, this line is going downhill, right, because the slope is negative, it forms a downhill right from left to right. And notice our other 2 which have positive slopes, they look like they're going uphill. Right? B and C both have this uphill tendency. But now let's look at one more thing here. Notice in B, our slope was 1 third and in C our slope is 2. Right? So 2 is quite a bit bigger than 1 third and look at how these lines look, right. In B, you kinda see like a soft growth here, right. It's kind of a little bit of an uphill where in C, where we have a slope of 2, it's a lot steeper. So the higher the slope is, the steeper it's gonna get this way. And if it were a really negative number, so if it were a negative 2, you could imagine it'd be a lot steeper going down. Cool? Alright. So that's how we calculate the slope. Let's move on.

Here’s what students ask on this topic:

What is the formula for calculating the slope of a line?

The formula for calculating the slope of a line is given by the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, it is expressed as:

$\frac{y_2-y_1}{x_2-x_1}$

Here, y2 and y1 are the y-coordinates of the two points, and x2 and x1 are the x-coordinates. This formula is often simplified to 'rise over run'.

How do you interpret a negative slope on a graph?

A negative slope on a graph indicates that as you move from left to right along the x-axis, the y-values decrease. This creates a downhill trend. In other words, for every unit increase in the x-value, the y-value decreases by the slope's magnitude. For example, a slope of -1 means that for every 1 unit increase in x, y decreases by 1 unit. This is often seen in graphs representing inverse relationships.

What does a slope of zero indicate about a line?

A slope of zero indicates that the line is horizontal. This means there is no vertical change as you move along the x-axis; the y-value remains constant. In mathematical terms, the rise is zero, so the slope formula $\frac{0}{x_2-x_1}$ simplifies to zero. A horizontal line represents a situation where the dependent variable does not change regardless of the independent variable.

How does the steepness of a slope relate to its numerical value?

The steepness of a slope is directly related to its numerical value. A larger absolute value of the slope indicates a steeper line. For example, a slope of 2 is steeper than a slope of 1/3. Positive slopes indicate an uphill trend, while negative slopes indicate a downhill trend. The greater the magnitude of the slope, the steeper the line appears on the graph. Conversely, smaller magnitudes result in gentler slopes.

How do you find the slope of a line given two points?

To find the slope of a line given two points, use the formula:

$\frac{y_2-y_1}{x_2-x_1}$

Identify the coordinates of the two points as (x₁, y₁) and (x₂, y₂). Subtract the y-coordinate of the first point from the y-coordinate of the second point to find the rise. Then, subtract the x-coordinate of the first point from the x-coordinate of the second point to find the run. Divide the rise by the run to get the slope.