Calculator
Example Data Table
| x | y | xy |
|---|---|---|
| 1 | 24 | 24 |
| 2 | 12 | 24 |
| 3 | 8 | 24 |
| 4 | 6 | 24 |
| 6 | 4 | 24 |
| 8 | 3 | 24 |
Formula Used
Inverse variation means one variable changes as the reciprocal of the other.
Core form: y = k / x
Equivalent form: x × y = k
Find the constant: k = x × y
Find y: y = k / x
Find x: x = k / y
Two pairs follow inverse variation when both products are equal.
How to Use This Calculator
- Select the calculation mode.
- Enter the known values in the visible fields.
- Click the Calculate button.
- Read the result table shown above the form.
- Review the plotted curve to inspect the relationship.
- Use the CSV button for spreadsheet export.
- Use the PDF button for a printable summary.
About Inverse Variation
Inverse variation describes a relationship where one quantity rises while the other falls, but their product stays constant. This pattern appears in many algebra, physics, finance, and geometry problems. When x doubles, y halves. When x triples, y becomes one third, provided the relationship is truly inverse.
This calculator supports several useful workflows. You can find the constant of variation from a known pair, solve a missing y value from k and x, solve a missing x value from k and y, complete a second pair, and verify whether two pairs belong to the same inverse rule. That makes it practical for homework, worksheets, checking examples, and quick classroom demonstrations.
The graph helps you inspect the shape of the relation. For positive constants, the curve appears in the first and third quadrants. For negative constants, the curve appears in the second and fourth quadrants. This visual pattern helps students understand why the function never touches x = 0.
The result table provides a direct check of the product x × y. If the product remains the same across valid pairs, the relationship fits inverse variation. If products differ, the values do not represent one shared inverse rule. The included example table gives a ready reference where the constant product stays fixed at 24.
Exports are included for convenience. CSV is useful for analysis in spreadsheet tools, while PDF gives a neat document for printing or sharing. Because all logic lives in one file, the page is simple to deploy and edit. You can expand the formulas, validation, or example content later without changing the overall structure.
FAQs
1. What is inverse variation?
Inverse variation means two variables have a constant product. As one increases, the other decreases proportionally. It is commonly written as y = k / x or xy = k.
2. How do I find the constant of variation?
Multiply any valid x and y pair. The product gives k. Once you know k, you can solve other missing values using the same inverse relationship.
3. Why can x not be zero?
Because y = k / x involves division by x. Division by zero is undefined, so inverse variation does not allow x = 0 when solving the formula.
4. Can this calculator verify two ordered pairs?
Yes. Enter both pairs in verification mode. If both products match, the pairs follow one inverse variation rule. If not, they do not match.
5. What does the graph show?
The graph shows the curve of y = k / x based on the computed constant. It helps you see how y changes as x moves away from zero.
6. When should I use the complete pair mode?
Use it when one full pair is known and you have either x2 or y2. The calculator fills the missing second value using the same constant.
7. What is the difference between direct and inverse variation?
Direct variation keeps y / x constant, while inverse variation keeps xy constant. In direct variation, values grow together. In inverse variation, one rises as the other falls.
8. Can I export my results?
Yes. The result block includes CSV and PDF download buttons. CSV works well for tables, while PDF is useful for printing and record keeping.