Rise — Definition, Formula & Examples

Rise is the vertical change (up or down) between two points on a line. It tells you how far a line goes up or down as you move from one point to another.

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ , the rise is the difference in their $y$ -coordinates, calculated as $y_2 - y_1$ . A positive rise indicates upward movement; a negative rise indicates downward movement.

Key Formula

\text{rise} = y_2 - y_1

Where:

$y_1$ = The y-coordinate of the first point
$y_2$ = The y-coordinate of the second point

How It Works

To find the rise, subtract the

y

-coordinate of your starting point from the

y

-coordinate of your ending point. You pair rise with run (the horizontal change) to calculate slope using the formula slope

= \frac{\text{rise}}{\text{run}}

. If the rise is positive, the line moves upward from left to right. If the rise is negative, the line moves downward.

Worked Example

Problem: Find the rise between the points (2, 3) and (5, 9).

Identify the y-coordinates: The first point has

y_1 = 3

and the second point has

y_2 = 9

Subtract: Subtract the first y-coordinate from the second.

\text{rise} = 9 - 3 = 6

Answer: The rise is 6, meaning the line goes up 6 units between these two points.

Why It Matters

Understanding rise is essential for calculating slope, which appears throughout algebra and geometry. Slope shows up in real contexts like measuring the steepness of a ramp, the grade of a road, or the rate of change in a graph of distance over time.

Common Mistakes

Mistake: Confusing rise with run by subtracting the x-coordinates instead of the y-coordinates.

Correction: Rise is always the change in

y

(vertical). Run is the change in

x

(horizontal). Remember: rise goes up-and-down, run goes side-to-side.