Slope

Slope measures a line’s vertical change per unit of horizontal change and represents a constant rate of change.

Slope compares vertical change with horizontal change. A line contains two marked points joined by a right-angled rise-and-run triangle. The vertical and horizontal legs make the signed ratio behind slope visible.

Slope is vertical change divided by horizontal change. On a line, it is the constant rate at which $y$ changes with respect to $x$.

Rise over run

For two distinct points $(x_1,y_1)$ and $(x_2,y_2)$,

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

The numerator is the rise and denominator is the run. The same direction must be used in both differences. Reversing both produces the same quotient; reversing only one changes the sign incorrectly.

Finding slope from two points

Check by choosing a smaller step: from $(-2,3)$, moving right $3$ should move up $6$, reaching $(1,9)$ on the same line.

Reading slope from a graph

Choose two clear grid-intersection points on the line. Draw a right-angle step between them, count vertical change with sign, and count horizontal change with sign.

A steep-looking graph does not automatically have a large slope: axis scales can distort appearance. Numerical rise and run are the evidence.

Types of slope

Positive slope: the line rises from left to right.
Negative slope: the line falls from left to right.
Zero slope: a horizontal line has no vertical change.
Undefined slope: a vertical line has zero horizontal change, so the slope would require division by zero.

The vertical line $x=4$ has undefined slope. The horizontal line $y=4$ has slope $0$.

Common mistakes

Mixing subtraction order. If the numerator uses second minus first, the denominator must do the same.

Calling vertical slope zero. Vertical lines have undefined slope; horizontal lines have zero slope.

Ignoring units and scales. Slope describes output units per input unit.

Taking the negative reciprocal for parallel lines. Equal slopes are parallel; negative reciprocals are perpendicular.

Quick self-check

Did I use two distinct points on the same line?
Are numerator and denominator differences in the same order?
Does the sign match the graph’s direction?
What units and contextual meaning belong to the slope?