The Significance of Negative Slope

In mathematics, the slope of a line (m) describes how rapidly or slowly change is occurring and in which direction, whether positive or negative. Linear functions—those whose graph is a straight line—have four possible types of slope: positive, negative, zero, and undefined. A function with a positive slope is represented by a line that goes up from left to right, while a function with a negative slope is represented by a line that goes down from left to right. A function with zero slope is represented by a horizontal line, and a function with an undefined slope is represented by a vertical line.

Slope is usually expressed as an absolute value. A positive value indicates a positive slope, while a negative value indicates a negative slope. In the function y = 3x, for example, the slope is positive 3, the coefficient of x.

In statistics, a graph with a negative slope represents a negative correlation between two variables. This means that as one variable increases, the other decreases and vice versa. Negative correlation represents a significant relationship between the variables x and y, which, depending on what they are modeling, can be understood as input and output, or cause and effect.

How to Find Slope

Negative slope is calculated just like any other type of slope. You can find it by dividing the rise of two points (the difference along the vertical or y-axis) by the run (the difference along the x-axis). Just remember that the "rise" is really a fall, so the resulting number will be negative. The formula for the slope can be expressed as follows:

m = (y2 - y1) / (x2 - x1)

Once you graph the line, you'll see that the slope is negative because the line goes down from left to right. Even without drawing a graph, you will be able to see that the slope is negative simply by calculating m using the values given for the two points. For example, suppose the slope of a line that contains the two points (2,-1) and (1,1) is:

m = [1 - (-1)] / (1 - 2)

m = (1 + 1) / -1

m = 2 / -1

m = -2

A slope of -2 means that for every positive change in x, there will be twice as much negative change in y.

Negative Slope = Negative Correlation

A negative slope demonstrates a negative correlation between the following:

• Variables 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. As the value of x increases, the value of y decreases. Likewise, as the value of x decreases, the value of y increases. Negative correlation, then, indicates a clear relationship between the variables, meaning one affects the other in a meaningful way.

In a scientific experiment, a negative correlation would show that an increase in the independent variable (the one manipulated by the researcher) would cause a decrease in the dependent variable (the one measured by the researcher). For example, a scientist might find that as predators are introduced into an environment, the number of prey gets smaller. In other words, there is a negative correlation between number of predators and number of prey.

Real-World Examples

A simple example of negative slope in the real world is going down a hill. The farther you travel, the farther down you drop. This can be represented as a mathematical function where x equals the distance traveled and y equals the elevation. Other examples of negative slope demonstrate the relationship between two variables might include:

Mr. Nguyen drinks caffeinated coffee two hours before his bedtime. The more cups of coffee he drinks (input), the fewer hours he will sleep (output).

Aisha is purchasing a plane ticket. The fewer days between the purchase date and the departure date (input), the more money Aisha will have to spend on airfare (output).

John is spending some of the money from his last paycheck on presents for his children. The more money John spends (input), the less money he will have in his bank account (output).

Mike has an exam at the end of the week. Unfortunately, he would rather spend his time watching sports on TV than studying for the test. The more time Mike spends watching TV (input), the lower Mike's score will be on the exam (output). (In contrast, the relationship between time spent studying and exam score would be represented by a positive correlation since an increase in studying would lead to a higher score.)