Algebra 2 / Unit 6: Rational Functions
6.1 Inverse Variation
Notes
Two quantities show inverse variation when their product is constant: $\displaystyle y = \frac{k}{x}$, or $xy = k$. The number $k$ is the constant of variation.
- As one quantity doubles, the other is cut in half.
- To solve: find $k$ from a known pair, then use it. Or use $x_1y_1 = x_2y_2$.
- Example: $y = 10$ when $x = 3$, so $k = 30$. When $x = 5$, $\displaystyle y = \frac{30}{5} = 6$.
- Compare with direct variation $y = kx$, where the ratio $\displaystyle \frac{y}{x}$ is constant.
Practice
$y$ varies inversely with $x$, and $y = 6$ when $x = 4$. Find $y$ when $x = 8$.
$k = 6 \cdot 4 = 24$, so $\displaystyle y = \frac{24}{8} = 3$.
Which equation shows inverse variation?
The product of $x$ and $y$ is constant.
The time to drive 240 miles varies inversely with speed. At 60 mph the trip takes 4 hours. How long does it take at 80 mph?
$k = 60 \cdot 4 = 240$, and $\displaystyle \frac{240}{80} = 3$.
Ready for more? Take the Unit 6 test.