Inverse Variation Calculator
Free Inverse variation Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs. Free to use with no signup required.
Formula
y = k / x^n
Where y is the dependent variable, k is the constant of variation (k = y * x^n), x is the independent variable, and n is the power (n=1 for simple inverse variation, n=2 for inverse square variation). The product y * x^n always equals the constant k.
Worked Examples
Example 1: Finding Unknown Value with Inverse Variation
Problem: If y varies inversely with x, and y = 12 when x = 5, find y when x = 15.
Solution: Step 1: Find k using y = k/x\nk = xy = 5 * 12 = 60\n\nStep 2: Write the equation\ny = 60/x\n\nStep 3: Substitute x = 15\ny = 60/15 = 4\n\nVerification: 5 * 12 = 60, 15 * 4 = 60 (constant product confirmed)
Result: k = 60 | y = 60/x | When x = 15, y = 4
Example 2: Inverse Square Variation in Physics
Problem: Light intensity varies inversely with the square of the distance. If intensity is 200 lumens at 3 meters, find intensity at 6 meters.
Solution: Step 1: Use y = k/x^2\nk = y * x^2 = 200 * 3^2 = 200 * 9 = 1800\n\nStep 2: Equation is I = 1800/d^2\n\nStep 3: At d = 6:\nI = 1800/6^2 = 1800/36 = 50 lumens\n\nDoubling the distance reduces intensity to 1/4 (200/4 = 50)
Result: k = 1800 | I = 1800/d^2 | At 6m: 50 lumens (25% of original)
Frequently Asked Questions
What is inverse variation and how does it differ from direct variation?
Inverse variation is a mathematical relationship where one variable increases as the other decreases, maintaining a constant product. If y varies inversely with x, the equation is y = k/x, where k is called the constant of variation. This contrasts with direct variation (y = kx) where both variables change in the same direction. In inverse variation, doubling x causes y to be halved, and tripling x reduces y to one-third. A real-world example is speed and travel time: if you double your speed, the travel time is cut in half. The product xy always equals the constant k, which provides a quick way to verify inverse variation in data sets.
How do you find the constant of variation k?
The constant of variation k is found by multiplying the known x and y values together. Since the inverse variation equation is y = k/x, rearranging gives k = xy. If you know that y = 12 when x = 3, then k = 3 times 12 = 36, and the complete equation is y = 36/x. Once k is established, you can find y for any x value by dividing k by x. For inverse variation with a power, y = k/x^n, the constant is k = y times x^n. The constant k represents the fixed product of the two varying quantities and its value determines the shape and position of the hyperbolic curve on the coordinate plane.
What does the graph of an inverse variation look like?
The graph of y = k/x is a rectangular hyperbola with two separate curves, one in the first quadrant and one in the third quadrant when k is positive. The graph never touches or crosses either axis because x cannot be zero (undefined division) and y can never actually reach zero. These axes serve as asymptotes that the curve approaches infinitely closely but never reaches. As x increases toward infinity, y approaches zero, and as x approaches zero from the positive side, y shoots toward infinity. The graph is symmetric about the origin, meaning rotating it 180 degrees about the origin produces the same curve. When k is negative, the hyperbola appears in the second and fourth quadrants instead.
What are real-world examples of inverse variation?
Inverse variation appears throughout science and everyday life. In physics, Boyle's Law states that gas pressure varies inversely with volume at constant temperature, so compressing a gas to half its volume doubles the pressure. In electrical circuits, Ohm's Law shows current varies inversely with resistance at constant voltage. Speed and travel time are inversely related for a fixed distance. The gravitational force between objects follows inverse square variation (y = k/x^2). In economics, if a fixed budget is divided among workers, each person receives less as more workers are added. Even photography uses inverse variation: aperture size and depth of field have an inverse relationship.
What is inverse square variation and how is it different?
Inverse square variation follows the formula y = k/x^2, meaning y varies inversely with the square of x. This produces a steeper decline than simple inverse variation. When x doubles, y becomes one-fourth of its original value rather than one-half. Inverse square variation governs many fundamental physical laws including gravitational attraction (Newton's Law of Gravity), electrostatic force (Coulomb's Law), light intensity from a point source, and sound intensity. The key difference is the rate of change: in regular inverse variation y = k/x, doubling x halves y; in inverse square variation y = k/x^2, doubling x reduces y to one-quarter. Inverse Variation Calculator supports both by allowing you to set the power parameter.
How do you determine if data follows an inverse variation pattern?
To test whether a data set represents inverse variation, multiply each x-y pair together. If the products xy are approximately constant, the data follows inverse variation y = k/x. For example, if your data pairs are (2, 15), (3, 10), (5, 6), and (6, 5), the products are 30, 30, 30, and 30, confirming inverse variation with k = 30. For inverse square variation, compute x^2 times y for each pair and check for a constant. If neither product is constant, the data may follow a different relationship. In practice, real-world data will not produce perfectly equal products due to measurement error, so look for products that are reasonably close to the same value.