Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 54

Chapter 4, Problem 54

What happens to y if y varies directly as x, and x is halved?

Verified step by step guidance

Understand the concept of direct variation: If y varies directly as x, it means that y is equal to a constant k multiplied by x. This can be written as the equation $y = k \times x$, where k is a constant.

Identify what happens when x is halved: If the original value of x is changed to $\frac{x}{2}$, substitute this new value into the direct variation equation.

Write the new equation for y when x is halved: $y_{new} = k \times \frac{x}{2}$.

Simplify the expression: $y_{new} = \frac{kx}{2}$.

Compare the new y value to the original y value: Since the original y was $y = kx$, the new y is exactly half of the original y, meaning y is halved when x is halved.

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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 a constant multiple of another, expressed as y = kx. If x changes, y changes proportionally, maintaining the constant ratio k.

Effect of Scaling the Independent Variable

When the independent variable x is scaled by a factor, the dependent variable y changes by the same factor in direct variation. For example, halving x results in y being halved as well.

Constant of Proportionality

The constant k in y = kx remains unchanged when x changes. It defines the fixed ratio between y and x, ensuring that y varies directly and proportionally with x.

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Related Practice

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (x - 1)/(x - 4) > 0

Textbook Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

Textbook Question

For each polynomial function, identify its graph from choices A–F.

$ƒ (x) = - (x - 2)^{2} (x - 5)$

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Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)²(x-5)

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (x + 1)/(x - 5) > 0

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Textbook Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

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