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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 45
Chapter 4, Problem 45

Describe in words the variation shown by the given equation. z=kxy2z = \(\frac{k\sqrt{x}\)}{y^2}

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Identify the variables and constants in the equation \(z = \frac{k \sqrt{x}}{y^2}\), where \(z\) is the dependent variable, \(x\) and \(y\) are independent variables, and \(k\) is a constant.
Recognize that \(z\) varies directly with the square root of \(x\), meaning as \(x\) increases, \(z\) increases proportionally to \(\sqrt{x}\).
Observe that \(z\) varies inversely with the square of \(y\), meaning as \(y\) increases, \(z\) decreases proportionally to \(\frac{1}{y^2}\).
Combine these observations to describe the overall variation: \(z\) increases with \(\sqrt{x}\) and decreases with \(y^2\).
Express the variation in words: \(z\) varies directly as the square root of \(x\) and inversely as the square of \(y\).

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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 = k√x / y², z varies directly with the square root of x, meaning as x increases, z increases proportionally to √x, assuming other variables remain constant.
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Inverse Variation

Inverse variation occurs when one variable increases as another decreases, typically expressed as a variable divided by another. Here, z varies inversely with y squared, indicating that as y increases, z decreases proportionally to 1/y², assuming other variables are constant.
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Square Root and Exponent Rules

Understanding square roots and exponents is essential to interpret the equation. The square root of x (√x) is equivalent to x raised to the 1/2 power, and y squared (y²) means y multiplied by itself. These operations affect how changes in x and y influence z.
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Related Practice
Textbook Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x4−2x3+x2+12x+8

Textbook Question

Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. g(x)=1/(x−1)

Textbook Question

Solve the equation 12x3+16x2−5x−3=0 given that -3/2 is a root.

Textbook Question

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)/(x+4)<0

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Textbook Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. x4−3x3−20x2−24x−8=0

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views
Textbook Question

Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x3−3x2−11x+6.