The slope of a line (m) describes how rapidly or slowly change is occurring.
Linear Functions have 4 types of slopes: positive, negative, zero, and undefined.
Negative Slope = Negative Correlation
A negative slope demonstrates a negative correlation between the following:
- x and y
- input and output
- independent variable and dependent variable
- cause and effect
Negative correlation occurs when the two variables of a function move in opposite directions.
Look at the linear function in the picture. As the values of x increase, the values of y decrease. Moving from left to right, trace the line with your finger. Notice how the line decreases.
Next, moving from right to left, trace the line with your finger. As the values of x decrease, the values of y increase. Notice how the line increases.
Real World Examples of Negative Slope
A simple example of negative slope is going down a hill. The further you travel, the further you drop.
Mr. Nguyen drinks caffeinated coffee two hours before his bed time. The more cups of coffee he drinks (input), the fewer hours he sleeps (output).
Aisha is purchasing a plane ticket. The fewer days between the purchase date and the departure date (input), the more money Aisha will spend on airfare (output).
Calculating Negative Slope
Negative slope is calculated just like any other type of slope. You can divide the rise of two points (vertical or y-axis) by the run (difference along the x-axis).
You just need to remember the "rise" is really a fall, so your number will be negative!
m = (y_{2} - y_{1}) / (x_{2} - x_{1})
If the line is graphed, you'll see the slope is negative because it will turn down (the left side will be higher than the right). If you're given two points that aren't graphed, you'll know the slope is negative because it will be a negative number.
For example, the slope of a line which contains points (2,-1) and (1,1) is:
m = [1 - (-1)] / (1 - 2)
m = (1 + 1) / -1
m = 2/-1
m = -2
Refer to the PDF, Calculate.Negative.Slope to learn how to use a graph and the slope formula to calculate a negative slope.
Edited by Anne Marie Helmenstine, Ph.D.