Understanding Slope

Algebra 1 · Axiom Academy

Slope is the rate of change of a line — the rise over the run — and its sign tells you which way the line leans. Pick any two points on a line. The rise is how far the line goes up or down between them; the run is how far it goes across. Slope, written m , is the rise divided by the run — the line's rate of change. rise over run, the change in y over the change in x A line with a positive slope goes up from left to right: as x increases, y increases too. The larger the slope, the steeper the climb. Hiking up a mountain, your elevation (the y -value) rises as you walk forward (the x -value). That upward rate of change is a positive slope. A line with a negative slope goes down from left to right: as x increases, y decreases. The more negative the slope, the steeper the downhill. Driving down a long mountain road, your altitude (the y -value) drops as your distance from the top (the x -value) grows. That is a negative rate of change. A horizontal line has a zero slope . There is no vertical change — the rise is 0 — no matter how far you run. Zero divided by any nonzero run is 0 , so the line is perfectly flat. Real-world: walking on flat ground On a level sidewalk your elevation never changes as you move forward. The rate of change of your height is exactly zero. A vertical line has an undefined slope . Here the run is 0 — there is no horizontal change at all. Because we can never divide by zero, the slope simply does not exist.

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