Desmos Activity: Exploring the Slope Formula

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.

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.