Concept · Graphing
What Is Slope? Rise Over Run Explained
Slope is one number that says two things: which way a line leans, and how sharply. Everything else about it follows from that.
What slope measures
The short answer to what is slope: it is how much a line rises for every one step you take to the right. One number, carrying both the direction of the lean and how steep it is.
A slope of 2 means the line climbs two units for every one unit across. A slope of −1/3 means it falls one unit for every three across. The sign is the direction; the size is the steepness.
Why it is rise over run
Slope is defined as the vertical change divided by the horizontal change between any two points on the line — rise over run.
- rise = 8 − 2 = 6the vertical change
- run = 4 − 1 = 3the horizontal change
- m = 6 ÷ 3 = 2the line climbs 2 for every 1 across
It is a ratio, which is why any two points on the same line give the same answer. Take a bigger step and both the rise and the run scale together, leaving the ratio untouched. That constancy is what makes a line a line.
The four cases
Every slope falls into one of four categories, and recognising them on sight saves a great deal of calculation.
| Slope | The line | Example |
|---|---|---|
| positive | rises from left to right | m = 2 |
| negative | falls from left to right | m = −0.5 |
| zero | horizontal — no rise at all | y = 3 |
| undefined | vertical — the run is zero | x = 4 |
The last two are the pair that gets confused. A horizontal line has slope zero, because the rise is zero and zero divided by anything is zero. A vertical line has undefined slope, because the run is zero and nothing can be divided by zero. Zero and undefined are not the same answer.
Slope as a rate of change
Away from graph paper, slope is a rate. If the horizontal axis is time and the vertical axis is distance, the slope is speed. If it is hours worked against pay, the slope is the hourly rate.
That reading is why slope matters far beyond straight lines. A steeper line means a faster rate; a negative slope means the quantity is falling. Reading a graph is largely reading its slope.
It is also where calculus begins. A curve has a different steepness at every point, and the derivative is the slope of the straight line that just touches it there.
Parallel and perpendicular
Because slope fixes direction completely, two facts follow immediately.
- Two lines are parallel when their slopes are equal. Same lean, so they never meet.
- Two lines are perpendicular when their slopes multiply to −1. A line of slope 2 is perpendicular to one of slope −1/2 — the negative reciprocal.
The perpendicular rule looks arbitrary until you picture it: turning a line through a right angle swaps the rise and the run, and flips one of their signs.
Where slope sits in y = mx + b
The equation y = mx + b names both numbers that fix a line. The m is the slope; the b is the y-intercept, the value of y when x is zero.
So the equation is a complete description: b says where the line starts on the vertical axis, and m says what it does from there. Two points, one slope calculation and one substitution give you both.
Questions about what is slope
What does slope mean in simple terms?
How much a line goes up for every one step to the right. A positive slope rises, a negative slope falls, and a bigger number means a steeper line.
What is the difference between zero slope and undefined slope?
A horizontal line has zero slope, because there is no rise. A vertical line has undefined slope, because the run is zero and division by zero has no value.
Does it matter which two points I pick?
No. Any two points on the same line give the same slope, because slope is a ratio and both the rise and the run scale together.
Is slope the same as gradient?
Yes. Gradient is the more common term in British usage and slope in American, but they mean the same thing for a straight line.