If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Lesson 2: Slope

# Intro to slope

Walk through a graphical explanation of how to find the slope from two points and what it means.
We can draw a line through any two points on the coordinate plane.
Let's take the points $\left(3,2\right)$ and $\left(5,8\right)$ as an example:
The slope of a line describes how steep a line is. Slope is the change in $y$ values divided by the change in $x$ values.
Let's find the slope of the line that goes through the points $\left(3,2\right)$ and $\left(5,8\right)$:
$\text{Slope}=\frac{\text{Change in y}}{\text{Change in x}}=\frac{6}{2}=3$
Use the graph below to find the slope of the line that goes through the points $\left(1,2\right)$ and $\left(6,6\right)$.
$\text{Slope}=$

Notice that both of the lines we've looked at so far have been increasing and have had positive slopes as a result. Now let's find the slope of a decreasing line.

## Negative slope

Let's find the slope of the line that goes through the points $\left(2,7\right)$ and $\left(5,1\right)$.
$\text{Slope}=\frac{\text{Change in y}}{\text{Change in x}}=\frac{-6}{3}=-2$
Wait a minute! Did you catch that? The change in $y$ values is negative because we went from $7$ down to $1$. This led to a negative slope, which makes sense because the line is decreasing.
Use the graph below to find the slope of the line that goes through the points $\left(1,9\right)$ and $\left(4,0\right)$.
$\text{Slope}=$

## Slope as "rise over run"

A lot of people remember slope as "rise over run" because slope is the "rise" (change in $y$) divided by the "run" (change in $x$).
$\text{Slope}=\frac{\text{Change in y}}{\text{Change in x}}=\frac{\text{Rise}}{\text{Run}}$

## Let's practice!

Heads up! All of the examples we've seen so far have been points in the first quadrant, but that won't always be the case in the practice problems.
1) Use the graph below to find the slope of the line that goes through the points $\left(7,4\right)$ and $\left(3,2\right)$.
$\text{Slope}=$

2) Use the graph below to find the slope of the line that goes through the points $\left(-6,9\right)$ and $\left(2,1\right)$.
$\text{Slope}=$

3) Use the graph below to find the slope of the line that goes through the points $\left(-8,-3\right)$ and $\left(4,-6\right)$.
$\text{Slope}=$

4) Use the graph below to find the slope of the line that goes through the points $\left(4,5\right)$ and $\left(9,5\right)$.
$\text{Slope}=$

5) Use the graph below to choose the slope of the line that goes through the points $\left(3,2\right)$ and $\left(3,8\right)$.
$\text{Slope}=$

## Challenge problems

See how well you understand slope by trying a couple of true/false problems.
6) A line with a slope of $5$ is steeper than a line with a slope of $\frac{1}{2}$

7) A line with a slope of $-5$ is steeper than a line with a slope of $-\frac{1}{2}$

## Want to join the conversation?

• why does math have to be so confusing?
• If you work hard then eventually math won't be as confusing!
• How can the slope value (1/2 or 5) be used in real life, and how can we use it in math?

Thanks!
• It could be used to simulate the steepness of a mountain/hill.
• Genuine question- when will i EVER use this IRL?
• no but you will for your math class so
• Can somebody tell me how to easily visualized. which slope is steeper?
• Try to think about it like this, imagine you are running up (positive slope) or down (negative slope) a flight of stairs.

If the slope is a larger number, than it is the same as taking several steps at once.
If the slope is a smaller number, it is as if you are taking less steps at once, not going up the flight of stairs as quickly.

Imagine a slope of 1, means for ever step you take with your feet you only go up one stair.

Imagine a slope of 5, this means for every step you take you go up 5 stairs. You get to the top and rise much quicker.

___________

Another visual example: Imagine you are going skiing. As you go down a slope, you expect the slope to be negative. You come from up high on the y axis and go down.

If now you go to a ski slope at -5 that means for every meter you glide forward on your skis towards to bottom of the hill (x-axis), you also go down 5 meters on the hills height (y-axis). This is very steep as you can imagine.

Now imagine you are going down a ski slope of only 1/2. This means for every 1 meter you glide forward on your ski you only get 1/2 meter further down the hills height.
• ma’am I do not get this🧍🏻‍♀️
• who are you talking to
• lmao what is y=mx+b gonna do for me in life
• y=mx+b aka "Slope Form" will help you financially as an example using this can help you see how much you've invested or even decreased and this helps the stock market to see how much progress their innovations are and how much interest their losing. hopes this helps. :]
(1 vote)
• how does this affect me
• You can use it to program. And to calculate landslides the steepness of hills or the angle of a roof.
• Is there another way of finding a slope without a graph?
• It should give you points on the graph, use the formula y2-y1/x2-x1, to find the slope, (y2 a y-coordinate, of one of the point, you are not multiplying anything)
Hope this helps :)
• So it doesn't matter then, what numbers you use for slope. What about when they don't give you a graph and make you find slope? I'm confused
• You haven't said what information you were given instead of a graph.

If you are given two points on the line, you can calculate the slope using the slope formula.

If you are given the equation of the line, you can:
-- Change the equation into slope-intercept form: y=mx+b and "m" will be the slope.
-- Or, you can calculate two points using the equation and then use the slope formula to calculate the slope.

All these options are covered in later videos.
• i dont understand the questions they asked me.🤷😭
• Try thinking about the formula of how to find the slope of a line.

Formula: Slope = The change in y divided by the change in x.

So let's say that you were trying to find the slope of a line based off the points (9, 5) and (5, 3).

For the first step, you would have to go from the point (9, 5) towards (5, 3) horizontally, AKA going across the x-intercept. That would be 4 units. Now, for the second step, you would have to do the same thing except vertically, AKA going down the y-intercept. That would be 2 units. So, if I plugged all of these numbers into the formula, this is what it would look like: 2/4 = ?

2 divided by 4 equals 0.5. Thus, the slope of this line would be 0.5.

If you still don't understand, you can always go through the articles and videos all over again.

Hope this helps in one way or another. :)

PS: It's done the same way if you do it on a negative slope. My trick for this is to turn the graph upside down and then make the slope a negative number.