# Desmos Activity: Exploring the Slope Formula

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Use a custom Desmos graphing calculator activity to explore the slope formula. The Desmos activity includes the slope formula equation. Students input the values for the two coordinates. They can also use the provided sliders to input the values. The Desmos interactive will calculate the slope of the line connecting the two points, as well as the graph of the line itself. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file.

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The following section includes background information on slope. This background also includes video resources and accompanying transcripts.

## What Is Slope?

Watch this video to learn about the slope formula. (The transcript is also included.)

## Slope Formula, Video 1

This video introduces you to the slope formula and how to use it. Find thousands of additional resources on Media4Math.com.

The video was uploaded on 10/15/2022.

You can view the video here.

The video lasts for 2 minutes and 24 seconds.

Video Transcript

This is the slope formula. Use it when you want to find the slope of the line that connects two points.

It is the ratio of the Rise over the Run, or the ratio of this vertical distance over this horizontal distance.

In terms of the coordinates, it is the ratio in the difference in the y coordinates over the difference in the x coordinates.

When you know the coordinates of two points, use this version of the slope formula.

Here are two points. This one has coordinates x1 and y1 and this one has coordinates x2 and y2.

Use those coordinates in the slope formula to calculate the slope.

Here are two points in Quadrant I. This point has coordinates 2, 3 and this point has coordinates 6,7.

To find the slope of the line that crosses these two points, use the slope formula. For the purposes of the slope formula, let's call these coordinates x1 and y1, and let's call these coordinates x2 and y 2.

Take those coordinates and plug them into the slope formula. y2 goes here and y1 goes here. x2 goes here and x1 goes here.

That means that 7 minus 3 in the numerator and 6 - 2 in the denominator.

Simplify the numerator and denominator.

Simplify the numerator and denominator. Then simplify the expression. The slope of the line that crosses these two points has a slope of 1.

Watch this video to learn about undefined slope. (The video transcript is also included.)

Video Transcript

Slope is defined as the ratio of the change in the y-coordinates over the change in the x-coordinates.

When calculating slopes, the ratio is one number divided by another number.

By definition a ratio cannot have a denominator of zero.

That's because division by zero is undefined.

In order for the slope ratio to have a zero denominator, that means that delta-x would have to equal zero.

What does "delta-x equals zero" correspond to?

Let's place two points on a coordinate grid, like this.

Delta-x is the horizontal separation between the two points.

If delta-x equals zero, that means there is no horizontal separation between the two points.

That means the two points are in this relative position.

Do you see what this means?

If you connect the points with a line, you'll see that this line is parallel with the y-axis.

Since the y-axis is a vertical axis, that means a line with an undefined slope is a vertical line.

It also means that each point has the same x-coordinate.

Let's look at an example.

Here are two points, A and B.

Point A has coordinates (1, 2).

Point B has coordinates (1, 8).

Notice that both points have the same x-coordinate, 1.

To find the slope of the line connecting these two points, use the slope formula, as shown here.

Plug in the values of the coordinates into the formula.

Because the x-coordinates are identical, the value for delta-x is zero.

But we know that a ratio cannot have zero in the denominator.

The fact that delta-x is zero means that the slope of the line is undefined.

So, what we've seen is that if two points have the same x-coordinate, then the line connecting them has an undefined slope.

Since the equation for any line can be written in slope-intercept form, as shown below, what happens when the slope is undefined?

With an undefined slope there is no equation in slope-intercept form.

In fact the equation of a vertical line is of the form x = c, for some constant c.

The graph of such a line is a vertical line.

Such a line does not define a function.

One of the ways to test if a graph is a function is if any two points share the same x-coordinate.

The vertical line test is used to determine if such graphs are functions.

## Slope Definition

Slope is a ratio. See the definition below.

Recall that a ratio can look like a fraction and can be thought of as one number divided by another. What two numbers make up this ratio? As you can see from the definition, it’s the ratio of the change in y-coordinates over the change in x-coordinates.

Where do these change in coordinates come from? Take a look at this example. There are two points, each with its own coordinates.

The difference in the y-coordinates is this:

The difference in the x-coordinates is this:

The ratio of these differences is shown below:

This ratio is also known as the slope formula.

### Undefined Slope

Because slope is a ratio, like all ratios, the denominator of the fraction form can’t be zero.

Look at this definition.