Inverse variation
Identify inverse variation relationships in Grade 10 algebra where y = k/x, find the constant of variation k from data points, and distinguish inverse from direct variation.
Key Concepts
If the product of two variables is a constant, the equation represents an inverse variation. The relationship is described by the equations: $$xy = k \text{ or } y = \frac{k}{x}$$, where k is a nonzero constant. As one variable increases, the other decreases, maintaining a constant product.
If a construction project requires 10 workers 12 days to complete, how long would it take 8 workers? The constant is $10 \cdot 12 = 120$ worker days. So, $8 \cdot d = 120 \implies d = 15$ days. For the data set with x values {2, 4, 8} and y values {16, 8, 4}, we check the products: $2 \cdot 16 = 32$, $4 \cdot 8 = 32$, and $8 \cdot 4 = 32$. This is an inverse variation with $k=32$. If a car travels at 50 miles per hour for 4 hours, the distance is 200 miles. If the speed changes to 40 miles per hour, the time becomes $t = \frac{200}{40} = 5$ hours.
Think of an inverse variation like sharing a pizza! The more friends (x) you invite, the smaller the slice (y) each person gets. But the total amount of pizza (k) stays the same. The product of friends and slice size is always one whole pizza. It’s a perfect see saw balance where one goes up, the other must go down.
Common Questions
What is the equation for inverse variation?
y = k/x, where k is the constant of variation. As x increases, y decreases proportionally. The product xy = k is always constant.
How do you find the constant of variation k from a table?
Compute xy for each data pair. If all products are equal, the data shows inverse variation and k equals that product. For example, if x=2,y=6, then k=12.
How do you distinguish inverse variation from direct variation?
Direct variation: y = kx (ratio y/x is constant). Inverse variation: y = k/x (product xy is constant). Plot the data: direct variation is linear through origin; inverse variation is a hyperbola.