Inverse variation
Inverse Variation
Inverse Variation is a topic in Rational Functions in the California Common Core State Standards. It is aligned to Standard A-CED.2, which requires students to create equations that describe relationships, including inverse variation.
Inverse variation \(y=\dfrac{k}{x}\) keeps the product \(xy=k\) constant; its graph is a hyperbola.
Theory
\(k\) is the constant of variation; the graph is a hyperbola.
The relationship:
How to solve variation problems
- Write \(y=\dfrac{k}{x}\).
- Substitute a known pair to find \(k\).
- Write the specific equation.
- Use it to find the missing value.
Use \(k=xy\).
| \(k\) | \(=\) | \((2)(6)=12\) |
| \(y\) | \(=\) | \(\dfrac{12}{x}\) |
Substitute.
| \(y\) | \(=\) | \(\dfrac{12}{4}=3\) |
Constant \(k=PV=500\).
| \(P\) | \(=\) | \(\dfrac{500}{10}=50\) |
Direct variation keeps \(\dfrac{y}{x}=k\) constant (a line); inverse keeps the product \(xy=k\) constant (a hyperbola).
Common pitfalls
Frequently asked questions
What is inverse variation?
A relationship \(y=\dfrac{k}{x}\) where the product \(xy\) is constant.
How do you find the constant of variation?
Multiply a known \(x\) and \(y\): \(k=xy\).
How is inverse variation different from direct?
Direct keeps the ratio \(y/x\) constant; inverse keeps the product \(xy\) constant.
What does the graph look like?
A hyperbola with the axes as asymptotes.