Problem 102 - College Algebra with Modeling and Visualization

Question

Assume that the constant of proportionality is positive. Suppose \(y\) is directly proportional to the second power of \(x .\) If \(x\) is halved, what happens to \(y ?\)

Step-by-Step Solution

Answer

If \(x\) is halved, \(y\) becomes a quarter of its original value.

Step 1: Understanding Direct Proportionality

When we say that a variable \( y \) is directly proportional to the second power of another variable \( x \), it means that \( y = kx^2 \), where \( k \) is the constant of proportionality. This tells us that as \( x \) changes, \( y \) changes according to the square of \( x \).

Step 2: Introducing the Halved Value of x

According to the exercise, if \( x \) is halved, it becomes \( \frac{x}{2} \). We will substitute this in the equation \( y = kx^2 \) to find out how \( y \) changes.

Step 3: Substitute the New Value of x

Substitute \( \frac{x}{2} \) into the equation: \[y = k \left( \frac{x}{2} \right)^2\] which simplifies to \[y = k \frac{x^2}{4}\]

Step 4: Compare New y with Original y

In the original equation, \( y = kx^2 \). For the new scenario, we have \( y = \frac{kx^2}{4} \). This shows that the new value of \( y \) is \( \frac{1}{4} \) of the original value since \( \frac{kx^2}{4} = \frac{1}{4}kx^2 \).

Key Concepts

Constant of Proportionality

The constant of proportionality is a crucial concept in understanding direct relationships between variables in mathematics. When a variable, such as \( y \), is said to be directly proportional to another variable, expressed typically as \( x \), there must be a constant factor involved. This constant is represented by \( k \) in the equation \( y = k x^2 \) in the case of direct proportionality to the second power of \( x \). Here’s what you need to know:

• **Defining the Constant**: \( k \) is a non-zero value that remains constant in variations of \( y \) with changes in \( x \). It essentially scales the relationship between these two variables.
• **Positive Values**: In the context of the exercise, \( k \) is assumed to be positive, which means as \( x \) increases or decreases, and \( y \) follows suit while being bound to grow or shrink proportional to this constant multiplier.

Whenever you analyze a proportionality scenario, think of \( k \) as a kind of 'proportion ruler' that defines how dramatically \( y \) will change to the alterations in \( x \). This understanding is pivotal when solving related mathematical problems.

Direct Variation

Direct variation indicates a direct relationship between variables where one variable changes directly with the square of another. In the problem, the equation \( y = k x^2 \) gives us a direct variation because \( y \) changes in response to the squared change in \( x \). Here are some key insights:

• **Linear Transformation**: Even though the relationship involves \( x^2 \) making it non-linear, it's still predictable due to its direct proportion nature, meaning there is a specific pattern or pathway that \( y \) will follow with changes to \( x \).
• **Halving the Input**: When \( x \) is halved, that means typing into this pattern, we halve not just \( x \) but also modify it squared, hence, \((\frac{x}{2})^2=\frac{x^2}{4}\).

This demonstrates how any modification to \( x \) systematically alters \( y \) by predetermined exponential steps, keeping the ratio constant between them, dictated by the previously mentioned \( k \).

Second Power Relationship

The second power relationship reveals the kind of proportionality between variables where one variable is proportional to the square of another. Such a relationship massively magnifies even minor shifts in the value of \( x \). Here's how it operates:

• **Understanding the Power of 2**: When \( x \) is squared (\( x^2 \)), it means each unit change in \( x \) is exponentially more significant, affecting \( y \) by the square of the change. This implies that halving \( x \) (as given \( \frac{x}{2} \)) quarters \( y \) (as seen with the result \( \frac{kx^2}{4} \)).
• **Influence on Outcomes**: Squaring \( x \) for \( y = kx^2 \) equation makes each alteration, especially reductions, in \( x \) have amplified consequences for \( y \). So you can expect noticeable changes even with small input modifications.

This insight helps reinforce the grasp over how variable interactions affect results, particularly highlighting why \( y \)'s outcome shrinks significantly when \( x \) is proportionally lowered.