Calculator
Example data table
| Type | k | x | y | Rule |
|---|---|---|---|---|
| Direct | 4 | 3 | 12 | y = 4x |
| Direct | 4 | 7 | 28 | y = 4x |
| Inverse | 24 | 3 | 8 | y = 24 / x |
| Inverse | 24 | 6 | 4 | y = 24 / x |
| Direct | 1.5 | 10 | 15 | y = 1.5x |
| Inverse | 30 | 5 | 6 | y = 30 / x |
Formula used
Direct variation
Direct variation means one value changes in the same ratio as the other. The model is y = kx. The constant is k = y / x when x is not zero.
Inverse variation
Inverse variation means one value increases while the other decreases proportionally. The model is y = k / x. The constant is k = xy when x is not zero.
Solving unknown values
For direct variation, solve y = kx or x = y / k. For inverse variation, solve y = k / x or x = k / y.
Checking a pair
In direct variation, each valid pair has the same ratio y / x. In inverse variation, each valid pair has the same product xy.
How to use this calculator
1. Choose the relationship
Select direct variation if y grows with x. Select inverse variation if y decreases as x increases.
2. Enter known values
You can type the constant k directly, or provide one complete pair. The calculator will derive the constant from the given data.
3. Add a second pair
Use the secondary fields to predict a missing value, compare another pair, or confirm the same variation model.
4. Configure the table and graph
Set the table start, step, and row count. You can also give custom graph limits for a focused visual range.
5. Review and export
After submission, the result appears above the form. Use the export buttons to download the summary and generated table.
About direct and inverse variation
Direct and inverse variation appear in algebra, geometry, physics, engineering, finance, and data analysis. They help describe how one quantity responds when another quantity changes under a fixed rule. In a direct relationship, the ratio between y and x stays constant. When x doubles, y doubles too. In an inverse relationship, the product of x and y stays constant. When x doubles, y becomes half as large.
This calculator supports both models in one place. You can enter the constant directly, derive it from a known pair, solve a missing value in a second pair, verify consistency, generate a working table, and inspect the curve visually. That makes it useful for homework, revision, quick checking, and demonstration during class or tutoring sessions.
The graph adds another layer of understanding. Direct variation produces a straight line through the origin when k is fixed. Inverse variation creates a curve with a break near zero because division by zero is undefined. The exported table and summary help you keep a record of your work for reports, notes, or practice sheets.
FAQs
1. What is direct variation?
Direct variation means y changes in the same proportion as x. The equation is y = kx. If x doubles, y also doubles when the constant k stays fixed.
2. What is inverse variation?
Inverse variation means y changes in the opposite proportional way to x. The equation is y = k / x. If x doubles, y becomes half as large.
3. How do I find the constant k?
For direct variation, divide y by x. For inverse variation, multiply x by y. The calculator can determine k automatically when one complete pair is available.
4. Can I solve a missing x-value?
Yes. Enter k and y for direct variation to solve x = y / k. For inverse variation, enter k and y to solve x = k / y.
5. Why is x = 0 invalid in inverse variation?
Inverse variation uses y = k / x. Division by zero is undefined, so x cannot be zero in that model. The graph also breaks near zero.
6. Why does the graph look different for each type?
Direct variation forms a straight line because y changes linearly with x. Inverse variation forms a curved hyperbola because y depends on the reciprocal of x.
7. What does the generated table show?
The generated table lists x-values from your chosen start and step. It calculates the matching y-values using the resolved constant and selected variation type.
8. Can I use this page for checking homework?
Yes. It is useful for verifying constants, solving missing values, comparing two pairs, and reviewing the graph and table before writing the final answer.