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

Verified step by step guidance

1. Identify the type of variation in the equation. The equation z = kx^2 √y suggests a combined variation where z varies directly with x squared and also directly with the square root of y.
2. Recognize the constant of variation, k, which scales the relationship between z, x, and y.
3. Understand the role of x^2 in the equation. As x increases or decreases, z is affected quadratically, meaning z increases by the square of x.
4. Analyze the role of √y in the equation. As y increases, the square root of y increases at a slower rate than y itself, but it still directly influences the increase in z.
5. Combine these observations to describe the variation: z varies directly with both x squared and the square root of y, scaled by the constant k.

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.