Problem 2 - Precalculus Mathematics for Calculus

Question

If the quantities \(x\) and \(y\) are related by the equation \(y=\frac{3}{x},\) then we say that \(y\) is _____ _____ to \(x\) and the constant of _____ is 3.

Step-by-Step Solution

Answer

y is inversely proportional to x, and the constant of proportionality is 3.

Step 1: Identify the Relationship Type

The given equation is \( y = \frac{3}{x} \). This form represents a type of relationship between \( y \) and \( x \) where one variable is inversely proportional to the other. This is because as \( x \) increases, \( y \) decreases and vice versa, maintaining a constant product.

Step 2: Determine the Correct Terminology

Since the relationship is such that \( y \) and \( x \) multiply to a constant (\( y \times x = 3 \)), \( y \) is described as 'inversely proportional' to \( x \). This is because the product of two inversely proportional quantities remains constant.

Step 3: Identify the Constant of Proportionality

In the equation \( y = \frac{3}{x} \), the constant associated with the inverse proportionality is the numerator of the fraction. Thus, the constant of proportionality is 3, as it is the factor that determines how \( y \) changes in response to \( x \).

Key Concepts

Proportionality Constant

When discussing inverse proportionality, a key concept is the "proportionality constant." It's the number that remains unchanged as two quantities change. Let's break this down: in the equation \( y = \frac{3}{x} \), 3 is called the proportionality constant. This constant tells us how much one quantity will affect the other in their inverse relationship. In simpler terms, it is the multiplier that connects the variables in an inverse setup.
Here's how it works:

• In the equation \( y = \frac{3}{x} \), for any value you choose for \( x \), multiplying \( y \) by \( x \) will always give you 3, which is our constant.
• This constant helps maintain balance in the inverse relationship between the variables.
• It helps us predict how one variable will change in relation to the other.

Understanding the proportionality constant is crucial for handling equations involving inverse relationships.

Inverse Relationship

An inverse relationship between two variables means that as one variable increases, the other decreases, and vice versa. Imagine a seesaw: when one side goes up, the other comes down. In the equation \( y = \frac{3}{x} \), this principle is clearly at play.
Here's a closer look:

• With inverse proportionality, the product of the two variables \( x \) and \( y \) is always constant. In our example, \( y \times x = 3 \).
• This means if you double \( x \), \( y \) must halve to keep the product same.

These inverse relationships are common in physics and everyday scenarios, like speed and travel time. The trick is understanding that one variable's rise comes at the cost of another's fall, maintaining the same overall product.

Variable Relationship

In mathematics, the relationship between variables can take different forms, such as direct or inverse. In the case of the equation \( y = \frac{3}{x} \), the relationship is inverse. Why? Because the variables \( x \) and \( y \) are connected in such a way that increases in one lead to decreases in the other.
Key features of variable relationships include:

• The specific equation form reveals the nature of their relationship. Here, \( y \) is dependent on \( x \) being in the denominator, forming an inverse relationship.
• Such a relationship is distinct from a direct proportionality, where both variables increase or decrease together.
• Recognizing the type of relationship helps in predicting and interpreting changes in the variables.

Grasping variable relationships aids in understanding how different elements in mathematical equations and real-world problems interact.