Slope is a number that describes how steep a line is and which direction it rises or falls. In geometry and algebra, it connects points on a coordinate plane to the equation of a line. A slope can represent real changes, such as speed, grade of a ramp, or cost per item.
Learning the slope formula helps you compare lines and understand graphs accurately.
To find slope from two points, measure the vertical change, called rise, and divide it by the horizontal change, called run. If the points are (x1, y1) and (x2, y2), the formula is m = (y2 - y1)/(x2 - x1). A positive slope rises from left to right, while a negative slope falls from left to right.
Zero and undefined slopes describe special horizontal and vertical lines.
Key Facts
- Slope formula: m = (y2 - y1)/(x2 - x1)
- Rise is the vertical change: rise = y2 - y1
- Run is the horizontal change: run = x2 - x1
- Positive slope means the line rises from left to right.
- Negative slope means the line falls from left to right.
- A horizontal line has slope 0, while a vertical line has undefined slope.
Vocabulary
- Slope
- Slope is the ratio of vertical change to horizontal change between any two points on a line.
- Rise
- Rise is the change in y-values between two points on a coordinate plane.
- Run
- Run is the change in x-values between two points on a coordinate plane.
- Coordinate Plane
- A coordinate plane is a grid formed by perpendicular x- and y-axes used to locate points.
- Undefined Slope
- Undefined slope occurs when a vertical line has zero run, so division by zero would be required.
Common Mistakes to Avoid
- Subtracting coordinates in different orders: Students may compute y2 - y1 but x1 - x2, which changes the sign of the slope incorrectly. Use the same point order in the numerator and denominator.
- Putting run over rise: Students sometimes write m = (x2 - x1)/(y2 - y1), which gives the reciprocal of the correct slope. Slope is always rise divided by run.
- Calling a vertical line's slope zero: A vertical line has no horizontal change, so the denominator is 0 and the slope is undefined. A slope of 0 belongs to a horizontal line.
- Ignoring negative signs: Students may drop a negative sign when subtracting coordinates, which changes whether the line rises or falls. Carefully subtract signed numbers before simplifying.
Practice Questions
- 1 Find the slope of the line through the points (2, 3) and (6, 11).
- 2 Find the slope of the line through the points (-4, 5) and (2, -1), then state whether the line rises or falls from left to right.
- 3 A line passes through two points with the same x-coordinate but different y-coordinates. Explain why its slope is undefined and describe what the line looks like on a graph.