Textbook Question
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)2+5
1105
views
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
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 11
Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z.
Verified step by step guidance1
Understand the phrase "x varies jointly as y and z" means that x is directly proportional to the product of y and z. This can be expressed as an equation involving a constant of proportionality, say k.
Write the joint variation equation as \(x = k \cdot y \cdot z\), where \(k\) is a constant.
To solve the equation for \(y\), isolate \(y\) on one side. Start by dividing both sides of the equation by \(kz\) (assuming \(k \neq 0\) and \(z \neq 0\)):
\[ y = \frac{x}{kz} \]
This equation expresses \(y\) in terms of \(x\), \(k\), and \(z\). If you have specific values for \(x\), \(k\), and \(z\), you can substitute them to find \(y\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
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 case, x varies jointly as y and z means x = kyz, where k is the constant of proportionality.
Formulating Equations from Word Problems
Translating a verbal relationship into an algebraic equation involves identifying variables and their relationships. Here, recognizing 'x varies jointly as y and z' helps write the equation x = kyz, setting the foundation for solving for a specific variable.
Recommended video:
05:56
Introduction to Rational Equations
Solving for a Variable
Solving for y means isolating y on one side of the equation. Starting from x = kyz, divide both sides by kz (assuming k and z are nonzero) to get y = x / (kz), which expresses y explicitly in terms of x, k, and z.
Recommended video:
Guided course
05:28Equations with Two Variables
Related Practice
Textbook Question
Use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).]
1576
views
Textbook Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As ______
865
views
Textbook Question
In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x3−2x2−11x+12
1081
views
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. 3x2+10x−8≤0
499
views
Textbook Question
Divide using long division. State the quotient, and the remainder, r(x). (4x4−4x2+6x)/(x−4)
692
views