Graph and Formula for Inverse Relationship

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In science and mathematics, an inverse relationship is a relationship between two variables in which the increase of one value causes the decrease of the other. Discover the definition of an inverse relationship and how to graph one.


What Is an Inverse Relationship?

Inverse relationships, also known as negative relationships, exist when two variables have a negative correlation. When one variable rises, another falls, and vice versa. These are inverse correlations; when one variable in the data set rises, the other falls. As a result, their ratios produce a negative number.

Inverse relationships shed light on cause and effect and aid businesses in forecasting trends. Understanding how one variable affects another can be useful in developing more efficient and cost-effective strategies.


How to Calculate an Inverse Relationship

When one variable increases, the other variable decreases in an inverse relationship. An inverse relationship is calculated using the formula: y = k ÷ x.

Mathematically, x and y represent the two variables, and k is a constant. As x increases, y decreases, and vice versa. For example, in a half marathon race, y represents time, k represents the distance, and x represents the runner's speed. If speed increases, the time to complete the race decreases.


What Is the Opposite of Inverse Relationship?

An inverse relationship is the inverse of a direct or proportional relationship. In a direct relationship, an increase in one variable causes an increase in another variable. On a graph, that would mean the x-values and y-values move up or down together. For example, the cost of manufacturing a car and its overall cost to the consumer may reflect a causal relationship: a higher manufacturing cost leads to a higher price tag for the buyer.

Variables move in opposite directions in inverse relationships, or as one moves up, the other moves down, and vice versa. Consider the relationship between plane travel time and customer comfort: as one variable (flight time) increases, the other variable (customer comfort) decreases.


3 Inverse Relationship Examples

Check below for examples of inverse relationships you might encounter in real life:

1. Hot weather and soup sales: People prefer soup in the cooler months, so soup companies will see a drop in sales as the temperature rises.

2. Interest rates and home purchases: Here, the first variable is interest rates (x); the second is the rate of home purchases (y). Low interest rates entice homebuyers, so when x increases, y decreases.

3. Unemployment rates and consumer spending: Higher unemployment means lower consumer spending. People with less money or less financial stability are more likely to save, spend less, and make purchases using coupons and discounts. In the opposite scenario, this would be a positive correlation: as employment rates rise, consumer spending may rise as well.


Inverse Relationship Graph

On a graph, an inverse relationship always has a negative slope. In general, at least three data points are required to reveal a negative relationship on a graph. Create a graph on which your x-axis represents one variable, and the y-axis represents the other. In this example, the x variable is soup sale revenue in thousands of dollars on a given day, and the y variable is the daily mean temperature on those days. Your variables might look like this:

X (1, 2, 3)
Y (50, 40, 30)

On a fifty-degree day, soup sales were at $1,000, but on a thirty-degree day, sales were at $3,000. On your chart, you will note a downward slope, indicating an inverse relationship: when temperatures go down, soup sales go up.

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