Textbook Question
In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function.
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 18
Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as z and the difference between y and w.
Verified step by step guidance1
Identify the phrase 'x varies jointly as z and the difference between y and w.' This means x is proportional to both z and (y - w) multiplied together.
Write the joint variation equation as: \(x = k \cdot z \cdot (y - w)\), where \(k\) is the constant of proportionality.
To solve for \(y\), start by isolating the term containing \(y\). Divide both sides of the equation by \(kz\): \(\frac{x}{kz} = y - w\).
Next, solve for \(y\) by adding \(w\) to both sides: \(y = \frac{x}{kz} + w\).
The equation is now expressed with \(y\) as the subject: \(y = \frac{x}{kz} + w\). This shows \(y\) in terms of \(x\), \(z\), \(w\), and the constant \(k\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Joint Variation
Joint variation describes a relationship where one variable varies directly as the product of two or more other variables. In this problem, x varies jointly as z and the difference between y and w, meaning x = k * z * (y - w) for some constant k.
Formulating Equations from Word Problems
Translating verbal descriptions into algebraic equations involves identifying variables and their relationships. Here, recognizing 'x varies jointly as z and (y - w)' helps set up the equation x = k * z * (y - w), which models the given relationship.
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Solving Equations for a Specific Variable
Solving for y means isolating y on one side of the equation. Starting from x = k * z * (y - w), you divide both sides by k*z and then add w to isolate y, resulting in y = (x / (k*z)) + w.
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Related Practice
Textbook Question
Show that f(x) = x^3 - 2x - 1 has a real zero between 1 and 2.
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Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 4x2 + 1 ≥ 4x
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Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation.
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Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the difference between y and w.
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Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation.
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