In Exercises 45–46, describe in words the variation shown by the given equation. z = kx^2 √y

Key Concepts

Direct Variation

Direct variation occurs when one variable is a constant multiple of another. In the context of the equation z = kx^2 √y, z varies directly with x^2 and √y, meaning that if x or y changes, z will change proportionally, provided k remains constant.

Constant of Variation

The constant of variation, represented by k in the equation, is a non-zero constant that defines the relationship between the variables. It determines how much z changes in response to changes in x and y, effectively scaling the relationship between these variables.

Square and Square Root Functions

In the equation, x is squared and y is under a square root, which affects how changes in these variables influence z. Squaring x amplifies its effect on z, while taking the square root of y moderates its influence, leading to a unique variation pattern that combines both exponential and radical behaviors.