Understanding Slope
Understanding Slope
The slope of a line measures how steep it is — how fast y changes as x changes. It is often described as "rise over run": the vertical change divided by the horizontal change between two points on the line.
The Slope Formula
Given two points (x_1, y_1) and (x_2, y_2) on a line, the slope m is:
m = \frac{y_2 - y_1}{x_2 - x_1}The numerator is the "rise" (change in y), and the denominator is the "run" (change in x). It doesn't matter which point you call (x_1,y_1) and which you call (x_2,y_2), as long as you're consistent about the order of subtraction in the top and bottom.
Example
Find the slope of the line through (1, 2) and (4, 11).
m = \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3A slope of 3 means that for every 1 unit you move to the right, the line rises 3 units.
The Four Types of Slope
- Positive slope: the line rises from left to right (as x increases, y increases).
- Negative slope: the line falls from left to right (as x increases, y decreases).
- Zero slope: the line is perfectly horizontal — y never changes, since y_2 - y_1 = 0 for any two points on it.
- Undefined slope: the line is perfectly vertical — the formula would require dividing by zero, since x_2 - x_1 = 0 for any two points on it.
Reading Slope from a Table of Points
Consider the points below, all lying on the same line:
x: 0, 1, 2, 3 y: -1, 1, 3, 5
Each time x increases by 1, y increases by 2. So the slope is m = \dfrac{2}{1} = 2. Being able to spot a constant rate of change in a table is the same idea as computing slope from two points.
Slope as a Rate of Change
Outside of pure algebra, slope shows up constantly as a real-world rate of change. If a car travels at a constant speed, plotting distance traveled against time gives a line whose slope is the car's speed (distance per unit time). If a tank is being filled at a steady rate, the slope of the volume-versus-time line is the fill rate. Whenever a quantity changes by the same fixed amount for every equal step of another quantity, that relationship is linear, and its slope tells you exactly how fast the change is happening.
Key Takeaways
- Slope is rise over run: m = \dfrac{y_2-y_1}{x_2-x_1}.
- Positive slope rises left to right; negative slope falls left to right.
- A horizontal line has slope 0; a vertical line has undefined slope.
- In a table of values, a constant slope shows up as a constant change in y per unit change in x.