Math101 · Independent Ontario learning library
Math101learn.math101.caSlope
Slope measures a line’s vertical change per unit of horizontal change and represents a constant rate of change.
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 $(x_2,y_2)$,
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 $0$.
Slope as a rate
In context, slope carries units. If cost $C$ is graphed against kilometres $d$, a slope of $1.80$ means $1.80$ dollars per kilometre. If water height is graphed against time, slope may be centimetres per minute.
The sign has meaning: a negative slope can represent cooling, decreasing balance, or descending elevation.
From an equation
In slope-intercept form,
the coefficient $m$ is slope. In standard form, rearrange or use coefficients. For $3x+2y=8$, solve for $y$: $2y=-3x+8$, so $y=-\tfrac32x+4$ and the slope is $-3/2$.
Parallel and perpendicular lines
Distinct parallel nonvertical lines have equal slopes. Perpendicular nonvertical lines have slopes whose product is $-1$, so each is the negative reciprocal of the other. A line with slope $2/3$ is perpendicular to one with slope $-3/2$.
Horizontal and vertical lines form a perpendicular pair even though the vertical slope is undefined.
Average rate of change
For a nonlinear graph, the slope between two points is an average rate of change over that interval. It need not equal the rate at every point. Calculus refines this idea into instantaneous rate of change.
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?
Related topics
Explore the idea
Function transformation
Change one quantity at a time and connect what moves to Slope.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the slope through (−2, 3) and (4, 15).
- m = (15 − 3)/(4 − (−2))
- = 12/6
- = 2
End of lesson
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