Problem 90 - A Graphical Approach to College Algebra

Question

Assume that the constant of variation is positive. Let \(y\) vary inversely with the second power of \(x .\) If \(x\) doubles, what happens to \(y ?\)

Step-by-Step Solution

Answer

When \( x \) doubles, \( y \) becomes one-fourth of its original value.

Step 1: Understand the Initial Relationship

Since "\( y \) varies inversely with the second power of \( x \)", we have the equation:\[ y = \frac{k}{x^2} \]where \( k \) is the constant of variation, which is positive.

Step 2: Express the Condition of Doubling \(x\)

When \( x \) doubles, the new value of \( x \) becomes \( 2x \). Substitute \( 2x \) into the equation:\[ y_{new} = \frac{k}{(2x)^2} \]

Step 3: Simplify the New Expression

Simplify the equation from Step 2:\[ y_{new} = \frac{k}{4x^2} \]This shows that when \( x \) doubles, \( y \) becomes \( \frac{1}{4} \) of its original value.

Key Concepts

Constant of Variation

In inverse variation problems, the **constant of variation** is a key element. It remains the same, regardless of the values that the variables take on. In the equation related to inverse variation, represented as \( y = \frac{k}{x^2} \), \( k \) denotes the constant of variation. For any changes in the values of \( x \) or \( y \), \( k \) stays constant, ensuring the relationship holds true. For instance, when \( x \) is doubled, \( y \) changes accordingly to retain the balance established by \( k \). Another important aspect of the constant of variation is that it is usually positive, which helps maintain the meaningful relationship between the variables in a physical context, like motion or physics problems. Understanding \( k \) is crucial to solving inverse variation problems, as it allows for the prediction of one variable given changes in another.

Inverse Proportionality

**Inverse proportionality** is a fascinating mathematical concept. It describes a relationship where one quantity increases, and another simultaneously decreases in such a way that the product of the two quantities remains constant. An inverse relationship, like in the equation \( y = \frac{k}{x^2} \), means that as \( x \) increases, \( y \) decreases and vice versa. When discussing inverse proportionality, it's essential to visualize how multiplying by the same factor in the denominator (like doubling \( x \)) results in division by the same factor squared for \( y \). Thus, if \( x \) becomes \( 2x \), then \( y \) becomes \( \frac{1}{4} \) of its original value. This balancing act exemplifies how inverse proportions work in real exchanges, such as speed versus time for a fixed journey or pressure versus volume in a given gas.

Power Functions

**Power functions** are a broad category of functions which generally take the form \( f(x) = kx^n \). These involve forms where variables are at the base and are raised to a certain power, or exponent, signified by \( n \). In the context of inverse variation, the exponent is negative, as shown through division in equations like \( y = \frac{k}{x^2} = kx^{-2} \). An understanding of power functions helps grasp inverse variation since the power rule dictates how variable changes affect an outcome. For instance, power functions help predict outcomes when a base number, like \( x \), undergoes doubling, tripling, or any other transformation. It's the squaring (or any other power) that dictates how "inversely" the dependent variable changes as compared to the independent variable, making power functions instrumental in exploring complex relationships between variables.