Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 46

Chapter 4, Problem 46

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

Verified step by step guidance

Identify the variables and constants in the equation

z = k x^{2} \sqrt{y}

, where

z

is the dependent variable,

x

and

y

are independent variables, and

k

is a constant of proportionality.

Recognize that

z

varies directly with the square of

x

, meaning if

x

increases,

z

increases proportionally to the square of

x

Understand that

z

also varies directly with the square root of

y

, so if

y

increases,

z

increases proportionally to the square root of

y

Combine these observations to describe the overall variation:

z

varies jointly as the square of

x

and the square root of

y

Summarize the variation in words:

z

increases as

x

increases squared and as the square root of

y

increases, scaled by the constant

k

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Direct Variation

Direct variation describes a relationship where one variable increases or decreases proportionally with another. In the equation z = kx^2 √y, z varies directly with x squared and the square root of y, meaning as x or y increase, z changes accordingly based on the constant k.

Exponents and Powers

Exponents indicate repeated multiplication of a base number. Here, x is raised to the power of 2 (x^2), meaning x is multiplied by itself. Understanding how exponents affect variables helps interpret how changes in x influence z.

Square Root Function

The square root function, denoted √y, represents a value that when squared returns y. It grows slower than linear functions, so as y increases, √y increases but at a decreasing rate. Recognizing this helps explain how y affects z in the equation.

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Related Practice

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Textbook Question

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Textbook Question

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