Variation models describe relationships with a constant multiplier. The words direct, inverse, and joint determine where the variables appear; a known data pair determines the constant \(k\).
Three core models
| Language | Model | Constant test |
|---|---|---|
| \(y\) varies directly as \(x\) | \(y=kx\) | \(y/x=k\) |
| \(y\) varies inversely as \(x\) | \(y=k/x\) or \(xy=k\) | \(xy=k\) |
| \(z\) varies jointly as \(x\) and \(y\) | \(z=kxy\) | \(z/(xy)=k\) |
| \(y\) varies directly as the square of \(x\) | \(y=kx^2\) | \(y/x^2=k\) |
| \(y\) varies inversely as the square of \(x\) | \(y=k/x^2\) | \(x^2y=k\) |
Find the constant of variation
The k workflow
- Translate
Write the correct model from the words.
- Substitute the known values
Use one matching set of variables to solve for \(k\).
- Rewrite the complete model
Put the value of \(k\) back into the variation equation.
- Answer and verify
Substitute the requested inputs and check the constant ratio or product.
Direct variation
If \(y\) varies directly as \(x\) and \(y=18\) when \(x=6\), find \(y\) when \(x=11\).
Use \(y=kx\): \(18=6k\), so \(k=3\).
The model is \(y=3x\).
At \(x=11\), \(y=33\).
Inverse variation
If \(p\) varies inversely as \(q\) and \(p=8\) when \(q=3\), find \(p\) when \(q=12\).
Use \(pq=k\): \(k=8(3)=24\).
Then \(p=24/q\).
At \(q=12\), \(p=2\).
Joint variation
If \(z\) varies jointly as \(x\) and \(y\), and \(z=30\) when \(x=3,y=5\), find \(z\) when \(x=4,y=7\).
\(30=k(3)(5)\), so \(k=2\).
Use \(z=2xy\).
Recognize variation in tables
| x | y in table A | y/x | y in table B | xy |
|---|---|---|---|---|
| \(1\) | \(4\) | \(4\) | \(24\) | \(24\) |
| \(2\) | \(8\) | \(4\) | \(12\) | \(24\) |
| \(4\) | \(16\) | \(4\) | \(6\) | \(24\) |
SAT strategy
Common mistakes and traps
- Treating \(y=mx+b\) with nonzero \(b\) as direct variation.
- Using \(y/x\) for inverse variation instead of \(xy\).
- Setting \(k=1\) without using the known pair.
- Ignoring the word square.
- Pairing values from different rows of a table.
Identify the model
If \(xy=42\) for every row, what kind of variation relates \(y\) and \(x\)?
Show answer and explanation
Answer: Inverse variation.
A constant product \(xy=k\) is equivalent to \(y=k/x\).
Key takeaways
- Direct: constant quotient and an origin-passing line.
- Inverse: constant product.
- Joint: multiply all named variables by \(k\).
- Find \(k\) from known data before predicting.
- Translate powers exactly as stated.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.