How to Use and Graph Slope-Intercept Form

Welcome to this how-to-slope-intercept-form guide! In this guide, we’ll look at how to use slope-intercept form and how to use that information when graphing! Buckle up, everyone! 🚀

The Equation

The Slope-Intercept Form 📈 looks like this: 

y=mx+b

Here, the m represents the slope, and b represents the y-intercept.

This form is used often because it’s really easy to graph! Because you’re given the y-intercept, you can think of it as (0,b) when looking at it as a coordinate.

This equation can be solved with just slope and/or y-intercept. Or, you’ll be given two points! 

How to Use the Point-Slope Form

If you’re given the slope and the y-intercept, you just need to plug each number in for their respective variables 🖩. That’s it! 

If you’re only given two points, you’ll have to find the slope of the line using the slope formula (m=y2-y1/x2-x1). Then, you’ll use one of the coordinates and plug it in for x and y. Once you do that, solve for b and then rewrite the equation, so the equation looks something like this:

y=3x+1

That’s all you need to do! If you need to graph it, plot the point (0,1) since 1 is the y-intercept. The y-intercept is the point where the line crosses the y-axis. Then, use the slope to plot another point, which could be either (-1,-2) or (1,4) depending on which direction you decide to go. 

But what does that mean? Let’s look at an example to visualize it! 🤩

Example

Find the equation of a line that has the slope of 3 and y-intercept of 1. Then graph the line. 

First, look at the equation that you need to use 👀: 

y=mx+b

If you have the right equation, you’re off to a good start ✔️. Let’s look at what information we have so far: we have the slope and y-intercept. Let's first plug in the slope for m: 

y=3x+b

Then plug in the y-intercept for b: 

y=3x+1

Yay! Wasn’t that so easy? That’s right: slope-intercept form is super easy to use and understand. The good news is that you’ll probably use it for the rest of your math career! So if you can, try to retain the information a little even after you take a test over this 😏 

Now let’s try to graph it. First, what is the y-intercept, or b? That’s right, 1 is the y-intercept.

Remember 🤔: when you’re graphing the y-intercept, the coordinate should look like (0,b). So the y-intercept in this problem should be (0,1). Let’s plot that point:

Image Courtesy of Desmos

Next, we know that the slope is 3. That means we rise 3 and run 1. Try graphing the slope: 

Image Courtesy of Desmos

If you did the rise over run correctly, you should have graphed either (-1,-2) or (1,4), depending on which way you decided to rise and run. Then, all you need to do is connect the dots!

Image Courtesy of Desmos

That’s it! You just graphed the line! Wasn't that so easy and fun? 😝

When slope and y-intercept are not given

Teachers will sometimes decide not to give you the y-intercept or the slope, and instead give you 2 points. All you need to do is use the slope formula, solve for b, and rewrite the equation into the form. 

Using the good old m = y2 - y1 / x2 - x1 formula, you’ll have to find the slope yourself. Plug those in for x and y in the equation, along with the slope you just found. That leaves b as the only variable in the equation. With that, just solve for b. Then, you’ll have both slope and y-intercept, and you can go with the normal processes!

Let’s look at an example:

Example 

Find the equation of a line that passes through (-2,-1) and (1,5).

Let's first find the slope with the two points given: 

m=y2-y1/x2-x1

With the slope formula, plug in the coordinates for the appropriate variable!

m=5-(-1)1-(-2)

When you subtract (or add, because of the negative signs), you should get this:

m=6/3

Since 6 is divisible by 3,

m=2

The slope equals to 2. 

So far, so good. Then plug it in for the equation. 

y=2x+b

Now, choose (out of the two) a point that you want to use. Here, (1,5) will be used. 

5=2(1)+b

Simplify if necessary: 

5=2+b

Subtract 2 from both sides:

3=b

Now you have y-intercept! Isn’t this so cool? Now that you have both slope and y-intercept, plug it back into the equation. 

y=2x+3

Tada! Even without slope and y-intercept, you still managed to break through and solve the problem! Well done! 🙌

If you graph it, it should look like this:

Image Courtesy of Desmos

Conclusion

Just like we said, using slope-intercept form is super fun and easy! Now that you know how to use it, you can have fun solving and graphing equations. 🤗