Inverse variation
Model and solve inverse variation problems in Grade 9 Algebra using y = k/x. Find the constant of variation k and use it to predict unknown values. (Saxon Algebra 1, Grade 9)
Key Concepts
Property An Inverse variation is a relationship between two variables whose product is a constant. The equation $xy = k$ or $y = \frac{k}{x}$, where $k$ is a nonzero constant, defines an inverse variation between $x$ and $y$. Explanation Think of it like a seesaw! As one variable goes up, the other must go down to keep their product, $k$, perfectly balanced and constant. Unlike direct variation where they rise and fall together, in an inverse relationship, they move in opposite directions to ensure that the value of $xy$ always stays the same magical number. Examples The relationship $xy = 15$ is an inverse variation because the product of $x$ and $y$ is a constant, 15. The relationship $y = 3x$ is a direct variation, not an inverse variation, because it's in the form $y=kx$. If 3 people take 8 hours to paint a fence, how long will it take 4 people? Here, the number of people and time vary inversely.
Common Questions
What is inverse variation in algebra?
Inverse variation describes a relationship where y = k/x for a nonzero constant k. As x increases, y decreases proportionally, and the product xy always equals the constant k. The graph is a hyperbola.
How do you find the constant of variation in an inverse variation?
Multiply any x-value by its corresponding y-value: k = xy. If this product is the same for every pair in the table, the relationship is an inverse variation and k is your constant.
How do you distinguish inverse variation from direct variation?
In direct variation y = kx the ratio y/x is constant, and the graph is a line through the origin. In inverse variation y = k/x the product xy is constant, and the graph is a hyperbola. Check which quantity stays constant to identify the type.