Reading a slope value
The table below summarizes what different slope values indicate about a line's direction.
| Slope value | Line behavior |
|---|---|
| Positive (m > 0) | Rises from left to right |
| Negative (m < 0) | Falls from left to right |
| Zero (m = 0) | Perfectly horizontal line |
| Undefined | Perfectly vertical line (x₁ = x₂) |
- A vertical line has an undefined slope because the horizontal change (x₂ − x₁) is zero, making the ratio undefined; the calculator instead reports the vertical line's equation directly as x = constant.
- Two identical points (x₁ = x₂ and y₁ = y₂) do not define a unique line, so no slope can be calculated.
- The steeper a line, the larger the magnitude of its slope; a slope of exactly 1 or −1 corresponds to a 45° angle of inclination.
What is slope?
Slope describes how steeply a line rises or falls, defined as the ratio of vertical change (rise) to horizontal change (run) between two points on the line: m = (y₂ − y₁) / (x₂ − x₁). A positive slope means the line rises left to right; a negative slope means it falls; a slope of zero means the line is perfectly horizontal.
This calculator also converts slope into an angle of inclination — the angle the line makes with the positive x-axis, calculated using the arctangent function — and writes the line's equation in slope-intercept form, y = mx + b.
How to use this slope calculator
- Enter the coordinates of the first point, x₁ and y₁.
- Enter the coordinates of the second point, x₂ and y₂.
- Read the slope, the angle of inclination in degrees, and the line's equation in slope-intercept form.
The slope formula
Slope is calculated first; the angle of inclination and equation are both derived from it.
Common mistakes
- Reversing the order of subtraction in the numerator and denominator (using y₁ − y₂ over x₂ − x₁), which flips the sign of the slope.
- Assuming a slope of zero and an undefined slope are the same thing — zero means horizontal, undefined means vertical.
- Confusing slope with angle of inclination — a slope of 1 corresponds to a 45° angle, not a slope of 45.
- Forgetting that slope is a ratio, not a distance — it has no units of length by itself.
Questions fréquentes
How do you calculate slope between two points?
Slope equals the change in y divided by the change in x: m = (y₂−y₁)/(x₂−x₁). For points (1,2) and (4,8), the slope is (8−2)/(4−1) = 6/3 = 2.
What does a slope of 2 mean?
A slope of 2 means the line rises 2 units vertically for every 1 unit it moves horizontally. It corresponds to an angle of inclination of arctan(2) ≈ 63.43° from the horizontal.
What is the equation of a line with slope 2 through (1,2)?
Using y = mx + b and solving for b: b = y₁ − m×x₁ = 2 − 2×1 = 0. The equation is y = 2x, which also passes through (4,8) since 2×4 = 8.
Why is the slope of a vertical line undefined?
A vertical line has the same x-coordinate for every point, making the denominator of the slope formula (x₂−x₁) equal to zero. Division by zero is undefined, so the slope cannot be expressed as a number — the line is instead described by the equation x = constant.
What is the difference between slope and angle of inclination?
Slope is the numeric ratio of rise to run (m); angle of inclination is the angle in degrees that the line makes with the positive x-axis, calculated as arctan(m). A slope of 1 gives an angle of 45°, but a slope of 2 gives an angle of about 63.43°, not 90°.
Can slope be negative?
Yes. A negative slope means the line falls as it moves from left to right. For example, points (0,4) and (2,0) give a slope of (0−4)/(2−0) = −2.
Références
- Weisstein, Eric W. "Slope." MathWorld — A Wolfram Web Resource.
- Standard coordinate-geometry and algebra textbook conventions (e.g. Larson, Precalculus, Cengage Learning).