Inverse Variation
In inverse variation, y = k/x: y is inversely proportional to x. The product xy = k is constant. Models include Boyle's law (Pโ1/V), speed-time, and resistance.
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Boyle's law: PโVโ = PโVโ (inverse variation). Speed ร time = distance; half speed doubles time. Resistance โ 1/area for conductors.
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Why: Inverse variation models real-world relationships: pressure-volume, speed-time, workers-days. The hyperbola y = k/x is the graph.
How: Use k = xy to find the constant. For y = k/x^n or y = k/(xz), extend the same logic with the appropriate variables.
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๐ Examples โ Click to Load
Solve Mode
y Values for Different x
Constant k Contribution
๐ Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
y = k/x โ xy = k (constant product)
Graph of y = k/x is a hyperbola
๐ Key Takeaways
- โข Inverse variation means y = k/x: y is inversely proportional to x โ as x increases, y decreases
- โข The constant of variation k = xy is the product of x and y โ it never changes along the hyperbola
- โข The graph of y = k/x is a hyperbola with asymptotes at x = 0 and y = 0
- โข Power inverse y = k/x^n: e.g., gravitational force F โ 1/rยฒ
- โข Joint inverse y = k/(xz): one variable varies inversely with the product of two others
๐ก Did You Know?
๐ How Inverse Variation Works
In y = k/x, k is the constant of proportionality. The product xy = k is always the same. If you double x, y halves. The graph is a hyperbola that approaches but never touches the axes.
Example: y = 20/x
If x = 2, then y = 10. The product 2 ร 10 = 20 is constant. For x = 4, y = 5. For x = 5, y = 4. As x grows, y shrinks.
Worked Example: Boyle's Law
At constant temperature, PV = k. If Pโ=2 atm, Vโ=10 L, then k = 20. For Vโ=5 L, Pโ = k/Vโ = 20/5 = 4 atm. Pressure doubles when volume halves.
Direct vs Inverse
Direct: y = kx (line through origin). Inverse: y = k/x (hyperbola). In direct variation, doubling x doubles y. In inverse, doubling x halves y.
๐ Table of Values (y = k/x, k = 20)
| x | y = 20/x | Product xy |
|---|---|---|
| 1 | 20 | 20 |
| 2 | 10 | 20 |
| 4 | 5 | 20 |
| 5 | 4 | 20 |
| 8 | 2.5 | 20 |
| 10 | 2 | 20 |
| 20 | 1 | 20 |
Notice: xy = 20 for every row. That is the defining property of inverse variation.
๐ Direct vs Inverse: Side-by-Side
Direct Variation y = kx
- โข Ratio y/x is constant
- โข Graph: straight line through origin
- โข Double x โ double y
- โข Examples: distance vs time, wages vs hours
Inverse Variation y = k/x
- โข Product xy is constant
- โข Graph: hyperbola with asymptotes
- โข Double x โ halve y
- โข Examples: pressure vs volume, speed vs time
๐ฏ Expert Tips
๐ก Product Test
For multiple points, compute xรy for each. If all products are equal, you have inverse variation.
๐ก Zero Warning
x and y can never be zero in y = k/x. The hyperbola has asymptotes at the axes.
๐ก Power Inverse
y = k/xยฒ: gravitational force. y = k/xยณ: intensity of light. The exponent changes the curve shape.
๐ก Compare with Direct
Same (x,y): direct gives k = y/x, inverse gives k = xy. They are fundamentally different relationships.
๐ฏ When to Use Inverse Variation
Use inverse variation when: (1) one quantity increases as another decreases, (2) the product of the two quantities is constant, (3) you see "inversely proportional" or "varies inversely." Examples: pressure-volume (Boyle), speed-time for fixed distance, workers-days for fixed task, resistance-current at constant voltage.
If doubling x halves y, you have inverse variation. If doubling x doubles y, you have direct variation.
๐ Reference: Inverse Variation Types
| Type | Formula |
|---|---|
| Simple | y = k/x |
| Power | y = k/x^n |
| Joint | y = k/(xz) |
| Finding k | k = xy (or yรx^n, yรxรz) |
โ FAQ
What is inverse variation?
Inverse variation means y = k/x: one variable is inversely proportional to the other. As x increases, y decreases. The product xy = k is constant.
What is the constant of variation?
The constant k in y = k/x. It equals xรy for any point on the curve. Also called the constant of proportionality.
How do I find k from a point?
Use k = x ร y. For example, if (2, 10) is on the curve, k = 2 ร 10 = 20.
What is power inverse variation?
y = k/x^n: y varies inversely with the nth power of x. Examples: gravity F โ 1/rยฒ, light intensity โ 1/rยฒ.
What is joint inverse variation?
y = k/(xz): y varies inversely with the product of x and z. Example: rate of work with multiple factors.
How does inverse differ from direct variation?
Direct: y = kx (line, doubling x doubles y). Inverse: y = k/x (hyperbola, doubling x halves y).
When is the graph in the third quadrant?
When k < 0. For y = k/x with k negative, x and y have opposite signs, so the curve lies in quadrants II and IV.
๐ข Quick Reference
๐ Hyperbolic Graph Characteristics
The graph of y = k/x (k > 0) has two branches: one in the first quadrant (x > 0, y > 0) and one in the third quadrant (x < 0, y < 0). The axes are asymptotes: the curve approaches but never touches x = 0 or y = 0. As x โ 0โบ, y โ โ. As x โ โ, y โ 0โบ.
For power inverse y = k/xยฒ, the curve falls off faster. Gravitational force and Coulomb force follow the inverse square law: doubling distance reduces force to 1/4.
โ ๏ธ Common Mistakes to Avoid
- Using k = y/x instead of k = xy for inverse variation (that is direct variation)
- Forgetting that x and y cannot be zero in y = k/x
- Confusing inverse (product constant) with direct (ratio constant)
- Using the wrong exponent in power inverse (e.g., F โ 1/rยฒ not 1/r)
๐ Summary
Inverse variation describes relationships where one variable increases as the other decreases. The key formula is y = k/x with k = xy. Use this calculator to find k from known (x,y), predict y for new x, or predict x for new y. Power inverse y = k/x^n and joint inverse y = k/(xz) extend the concept for physics and multi-factor problems.
Switch between Simple and Advanced modes to access power and joint inverse variation. The Bar chart shows y for sample x values; the Doughnut shows k's contribution. Click examples to load real-world scenarios.
โ ๏ธ Disclaimer: This calculator assumes ideal inverse, power inverse, or joint inverse variation. Real-world data may have measurement error or not follow these models exactly. Educational use only.
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