Owlgebra

Understanding Slope: Rise over Run with a Graph

Slope describes how much y changes for each 1-unit increase in x on a straight line. On a graph, x is horizontal and y is vertical. A point (x, y) gives both coordinates.

The Big Idea

Slope = change in y ÷ change in x. This is often called rise over run. Use the same order for both subtractions, and keep track of signs.

Work Through an Example

Find the slope of the line through A = (1, 2) and B = (4, 8).

A line with slope 2 through (1, 2) and (4, 8)From A at x=1, y=2, move right 3 units to x=4, then up 6 units to B at y=8. Slope is 6 divided by 3, or 2. The horizontal and vertical drawing scales differ; use the labeled units.012345246810xyA (1, 2)B (4, 8)run = 3rise = 6
Rise 6, run 3: y increases by 2 for every 1-unit increase in x. Read the axis labels; the picture uses different horizontal and vertical scales.

  1. Find the vertical change: 8 − 2 = 6.

    Why: Going from A to B, y increases from 2 to 8. That is a rise of 6 units.

  2. Find the horizontal change: 4 − 1 = 3.

    Why: In the same direction, x increases from 1 to 4. That is a run of 3 units.

  3. Divide: slope = = 2.

    Why: A rise of 6 spread across a run of 3 means y increases by 2 for each 1-unit increase in x.

  4. Check the direction and scale.

    Why: The line goes up as you move right, so a positive slope makes sense. From (1, 2), moving right 3 and up 6 reaches (4, 8).

A Common Mix-Up

Do not mix point order: would have the wrong sign. Reversing BOTH subtractions is fine: = = 2.

Try Another Way to Picture It

Picture a staircase that rises 2 units every time you travel 1 unit to the right. The line follows that same rate. A line going down to the right has negative slope. A horizontal line has slope 0. A vertical line has undefined slope because its run is 0 and division by zero is undefined.

Your Turn

Try these on paper, then open each answer to check your reasoning. These examples do not record a score or change your learning progress.

Find the slope through (0, 5) and (2, 1).

Show answer and explanation

−2

The rise is 1 − 5 = −4 and the run is 2 − 0 = 2. Slope = = −2.

What is the slope through (−1, 3) and (4, 3)?

Show answer and explanation

0

The y-values are equal, so the rise is 0. The run is 5. Slope = = 0: a horizontal line.

Keep This in Mind

Subtract the y-values, subtract the x-values in the same order, and divide. Check the graph to see whether the sign makes sense.