Understanding Slope

Slope is simply the ratio of vertical change to horizontal change in a line. Think of it as measuring how steep a hill is—the steeper the hill, the larger the slope value. Mathematically, we express slope using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x} = \frac{Rise}{Run}$

When finding slope from a graph, remember some simple rules: moving UP is positive rise, DOWN is negative; moving RIGHT is positive run, LEFT is negative. You just need to pick any two points on the line and apply the formula. For example, if you move up 2 units and right 3 units, the slope is $\frac{2}{3}$.

Working with coordinate points requires the same approach. For points $3, -5$ and $-7, 2$, we calculate: $m = \frac{2 - (-5)}{-7 - 3} = \frac{7}{-10} = -\frac{7}{10}$. This gives us a negative slope, indicating the line falls from left to right.

Try This! Look around you for real-world slopes—stairs, ramps, roofs. Estimate which ones have steeper slopes (larger values) and which have gentler slopes (smaller values).

Slopes come in four main types: positive (line rises from left to right), negative (line falls from left to right), zero (horizontal line), and undefined (vertical line). Each type tells you something important about the relationship between the variables the line represents.