Textbook Question
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial 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 20
Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the sum of y and w.
Verified step by step guidance1
Identify the type of variation described: "x varies directly as z" means x is proportional to z, so we can write \(x = k \cdot z\) for some constant \(k\).
The phrase "and inversely as the sum of y and w" means x is inversely proportional to \((y + w)\), so we include this in the equation as \(x = \frac{k \cdot z}{y + w}\).
Write the full equation expressing the relationship: \(x = \frac{k \cdot z}{y + w}\), where \(k\) is the constant of proportionality.
To solve for \(y\), start by multiplying both sides of the equation by \((y + w)\) to eliminate the denominator: \(x(y + w) = k \cdot z\).
Next, divide both sides by \(x\) to isolate \((y + w)\): \(y + w = \frac{k \cdot z}{x}\). Finally, subtract \(w\) from both sides to solve for \(y\): \(y = \frac{k \cdot z}{x} - w\).
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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 is proportional to another. If x varies directly as z, it means x = k * z for some constant k. This implies that as z increases, x increases proportionally, and vice versa.
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Maximum Turning Points of a Polynomial Function
Inverse Variation
Inverse variation means one variable changes in the opposite way to another. If x varies inversely as a quantity, then x = k / (that quantity). Here, as the denominator increases, x decreases, showing an inverse proportionality.
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Forming and Solving Equations for a Variable
To solve for y, first write the equation expressing the given variation relationships. Then, isolate y by algebraic manipulation, such as multiplying both sides, combining like terms, and using inverse operations to express y explicitly.
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Related Practice
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
In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. 6x3+25x2−24x+5=0
Textbook Question
Divide using synthetic division. (4x3−3x2+3x−1)÷(x−1)
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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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Textbook Question
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function.
3
views