Slope -

Introduction to slope

Slope is a number that tells us how steep a line is. Here are some slope values:

This video has more details.

Mathisfun article on slope: https://www.mathsisfun.com/geometry/slope.html

Calculating Slope using Two Points

The formula for finding the slope of a straight line through two known points is:

$m=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}$

Let the left point be $(x_1,y_1)$ , and the right point be $(x_2,y_2)$ . (You can switch them but don’t mix them).

Example: Find the slope of the line that goes through the points $(1,8)$ and $(7,50)$

Let’s label the coordinates with the letters we use in the formula:

Now substitute those values into the formula:

(1) $\begin{align*}m&=\frac{y_2-y_1}{x_2-x_1}\\[10pt]&=\frac{50-8}{7-1}=\frac{42}{6}=7\end{align*}$

Another way is to find the rise and run by drawing a right triangle on the graph:

Either way, we now know that for every one horizontal increase, the $y$ coordinate increases by 7:

Practice the technique here:

link

Slope is a number that represents how quickly a line segment goes up a grid. Slope ‘8’ means that it goes up 8 vertical units for every one horizontal unit. The grid may or may not be drawn. What matters are the coordinates.

Move the line segment to create a certain slope

When a line or line segment is drawn on a 1:1 grid, we may simply count boxes to calculate slope:

$\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{\text{boxes up/down}}{\text{boxes along}}$

Use the applet to create line segments of the following slopes:

Slope AB = 5
SlopeAB = -3
Slope AB = 0
Slope AB = 1
Slope AB = -1
Infinitely steep
Slope AB = $\frac{5}{2}$
Slope AB = $-\frac{1}{4}$

Try this quiz:

Parallel and Perpendicular

Parallel lines have the same slope.

Perpendicular lines meet at 90 degrees.

We can make a second line perpendicular to another line using the slope. If the first line (or segment) has slope $\frac{3}{4}$ , a perpendicular line will have slope $-\frac{4}{3}$ . Try it out: