Slope measures how steep a line is and which direction it moves as you go from left to right. It is one of the most important ideas in algebra because it connects graphs, equations, tables, and real-world rates of change. On a coordinate plane, slope compares the vertical change to the horizontal change between two points.
This is why slope is often remembered as rise over run.
To find slope, choose two points on a line and subtract their coordinates in the same order. The formula m = (y2 - y1) / (x2 - x1) gives the slope from any two points, as long as x2 and x1 are not equal. In the equation y = mx + b, the number m is the slope and b is the y-intercept.
Positive, negative, zero, and undefined slopes each create a different kind of line on the graph.
Key Facts
- Slope = rise / run
- m = (y2 - y1) / (x2 - x1)
- In y = mx + b, m is the slope and b is the y-intercept.
- 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, and a vertical line has undefined slope.
Vocabulary
- Slope
- Slope is the ratio of vertical change to horizontal change for a line.
- Rise
- Rise is the vertical change between two points on a graph.
- Run
- Run is the horizontal change between two points on a graph.
- Y-intercept
- The y-intercept is the point where a line crosses the y-axis.
- Undefined slope
- Undefined slope occurs when a line is vertical and the run is 0.
Common Mistakes to Avoid
- Subtracting coordinates in different orders is wrong because the rise and run must match the same point order. If you use y2 - y1, you must also use x2 - x1.
- Writing slope as run over rise is wrong because slope is vertical change divided by horizontal change. Always use m = rise / run.
- Calling a vertical line slope 0 is wrong because a vertical line has run = 0, which would require division by zero. Vertical lines have undefined slope.
- Ignoring the sign of the slope is wrong because the sign tells the direction of the line. A line that falls from left to right has negative slope.
Practice Questions
- 1 Find the slope of the line through the points (2, 3) and (6, 11).
- 2 A line has equation y = -3x + 7. What is its slope, and what is its y-intercept?
- 3 Explain how you can tell from a graph whether a line has positive slope, negative slope, zero slope, or undefined slope.