Introduction to Slope

Have you ever walked up a steep hill or ridden down a ramp? The "steepness" of that hill or ramp is what we call slope in mathematics. In the coordinate plane, slope measures how fast a straight line goes up or down.

What is Slope?

Slope is the ratio of the vertical change to the horizontal change between any two points on a line. We often refer to this as rise over run.

$\text{Slope} = \frac{\text{Rise}}{\text{Run}}$

• Rise: The change in the $y$-direction (how far up or down you go).
• Run: The change in the $x$-direction (how far left or right you go).

The Slope Formula

If you know the exact coordinates of two points on a line, $(x_1, y_1)$ and $(x_2, y_2)$, you can calculate the slope (usually represented by the letter $m$) using this formula:

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

Types of Slope

By reading a graph from left to right, you can quickly determine what kind of slope a line has:

• Positive Slope: The line goes upward. As $x$ increases, $y$ increases.
• Negative Slope: The line goes downward. As $x$ increases, $y$ decreases.
• Zero Slope: A perfectly flat, horizontal line. There is no vertical change (the rise is $0$).
• Undefined Slope: A perfectly straight vertical line. There is no horizontal change (the run is $0$, and you cannot divide by zero).

Example Problems

Example 1: Find the slope of the line passing through $(1, 2)$ and $(3, 8)$.

1. Identify the points: Let $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (3, 8)$.
2. Apply the slope formula: $m = \frac{8 - 2}{3 - 1}$ $m = \frac{6}{2} = 3$ The slope is $3$. This means for every $1$ unit you move to the right, the line rises $3$ units.

Example 2: Determining the sign of a slope from a graph. If you look at a graph and the line is pointing down and to the right, the slope is negative. If it points up and to the right, it is positive.

Example 3: Finding rate of change from a table. Suppose you have a table tracking a constant rate of change:

$x$	$y$
$0$	$4$
$2$	$10$
$4$	$16$

Pick any two pairs of points, such as $(0, 4)$ and $(2, 10)$, and use the slope formula: $\text{Rate of Change} = \frac{10 - 4}{2 - 0} = \frac{6}{2} = 3$ The rate of change (which is the slope) is $3$.