Problem 88 - A Graphical Approach to Precalculus with Limits

Question

For \(k>0,\) if \(y\) varies inversely with \(x,\) when \(x\) increases, \(y\) _____, and when \(x\) decreases, \(y\) _________.

Step-by-Step Solution

Answer

y decreases when x increases; y increases when x decreases.

Step 1: Understand Inverse Variation

When two variables vary inversely, it means their product is constant. In this exercise, it means that as one variable increases, the other variable must decrease to keep the product constant.

Step 2: Mathematical Representation

When two variables vary inversely, we can express it with the equation: \( y = \frac{k}{x} \) where \( k \) is a positive constant.

Step 3: Analyze Effect of Increasing x

If \( x \) increases in the equation \( y = \frac{k}{x} \), the denominator becomes larger, resulting in a smaller value for \( y \). Thus, when \( x \) increases, \( y \) decreases.

Step 4: Analyze Effect of Decreasing x

Conversely, if \( x \) decreases, the denominator becomes smaller, meaning \( y \) becomes larger. Therefore, when \( x \) decreases, \( y \) increases.

Key Concepts

Mathematical Representation

In inverse variation, understanding the mathematical representation helps us grasp the nature of the relationship between two variables. When we say that a variable \( y \) varies inversely with another variable \( x \), it means that as one variable increases, the other one must decrease to maintain a balance.

This can be expressed through the equation \( y = \frac{k}{x} \), where \( k \) is a positive constant, often referred to as the constant of variation. This formula effectively shows that the product of \( y \) and \( x \) is equal to \( k \), emphasizing their inverse relationship.

By plugging different values into this formula, you see how an increase in \( x \) directly affects \( y \), based on the value of \( k \). This constant gives us a tool to calculate either variable if one is missing.

Constant Product

A key concept in the understanding of inverse variation is the idea of a constant product. In the context of inverse relationships, the product of two variables stays constant. This means if \( y \) is inversely proportional to \( x \), then the product \( x \times y = k \), where \( k \) remains constant.

Essentially, no matter how values of \( x \) and \( y \) change, as long as they adhere to the inverse relationship, the result of multiplying them will always be the same constant \( k \).

This helps keep the relationship between the variables stable and predictable, allowing us to determine one variable given the other. For instance, if \( x \) is doubled, the value of \( y \) will be halved, keeping their product equal to \( k \).

Variable Relationship

Inverse variation centers on the intricate relationship between two changing variables. This relationship is defined by oppositional movements—when one variable increases, the other decreases.

Perhaps one of the most intuitive ways to understand this is to think about gears in a machine. When one gear turns faster, the other connected gear slows down proportionally. Similarly, in inverse variation, an increase in \( x \) results in a decrease in \( y \) and vice versa.

This type of relationship is frequently observed in real-world applications such as speed and travel time: as you increase speed (\( x \)), the travel time (\( y \)) to your destination decreases.

Proportionality

While direct proportionality is a more straightforward concept, inverse proportionality requires an understanding of how differences in one variable are balanced by changes in another.

In inverse proportion, even though the variables are not directly linked, their relationship remains tightly regulated by the constant \( k \). This variance keeps the variables' interactions consistent and predictable.

For example, when you understand that sound frequency and wavelength are inversely proportional, it means increasing frequency results in a decrease in wavelength. This principle ensures that even in seemingly complex relationships, there's an underlying proportional balance maintained.