Problem 92 - A Graphical Approach to Precalculus with Limits

Question

Assume that the constant of variation 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

\( y \) becomes one quarter of its original value when \( x \) is halved.

Step 1: Understanding Direct Proportionality

When a quantity is directly proportional to the square of another, we use the expression \( y = kx^2 \), where \( k \) is a positive constant of variation. This means \( y \) changes as the square of \( x \) changes.

Step 2: Setting Up the Original Equation

Initially, assume a specific value of \( x = a \). Then, \( y = k(a^2) \). Let's consider this as our base comparison state, i.e., \( y_1 = ka^2 \).

Step 3: Considering the Halved Value of x

If \( x \) is halved, then \( x = \frac{a}{2} \). We substitute this new value into the proportionality equation: \( y_2 = k \left(\frac{a}{2}\right)^2 \).

Step 4: Simplifying the New Expression

We simplify the new equation: \( y_2 = k \left(\frac{a}{2}\right)^2 = k \left( \frac{a^2}{4} \right) = \frac{k a^2}{4} \).

Step 5: Comparing Original and New Values of y

Originally, \( y_1 = ka^2 \). Now \( y_2 = \frac{ka^2}{4} \). Comparing these two expressions, \( y_2 = \frac{1}{4} y_1 \). This means \( y \) becomes one quarter of its original value when \( x \) is halved.

Key Concepts

Constant of Variation

The constant of variation is a crucial factor in mathematical relationships that exhibit direct proportionality. When two variables are directly proportional, there exists a constant, often denoted by \( k \), that relates these two variables to each other.
In terms of writing, if a variable \( y \) is directly proportional to the square of another variable \( x \), this relationship can be described by the equation \( y = kx^2 \).
Here, \( k \) is what we refer to as the "constant of variation." It is assumed to be positive to ensure a straightforward and meaningful relationship.

• \( k \) determines how much \( y \) will change as \( x \) changes. A larger \( k \) implies a more significant change in \( y \) for the same change in \( x \).
• This constant remains the same once defined, as long as the relationship between \( y \) and \( x^2 \) remains unchanged.

Understanding the constant of variation makes it easier to predict outcomes in scenarios involving proportionality.

Square of a Variable

The concept of squaring a variable is vital when discussing relationships like direct proportionality involving squares. When a variable, say \( x \), is squared, it means it is multiplied by itself; mathematically, this is written as \( x^2 \).
Squaring has unique properties:

• Squaring a positive number results in a positive outcome, while squaring a negative number also yields a positive result because a negative multiplied by a negative gives a positive.
• The square of zero remains zero.

Now, if \( y \) depends on \( x^2 \), any changes to \( x \) will have a squared effect on \( y \). For instance, if \( x \) is halved, the new value \( y_2 \) can be calculated by replacing \( x \) in the equation \( y = kx^2 \) with \( \frac{x}{2} \) and squaring it, resulting in the value being divided by 4.

Proportionality Equation

A proportionality equation like \( y = kx^2 \) is a mathematical expression that clearly defines how two variables are connected through direct proportionality. In this case, \( y \) is directly proportional to the square of \( x \), with \( k \) being the constant of variation.
This equation conveys the direct relationship between \( y \) and \( x \):

• If \( x \) increases, \( y \) increases by the square of that increase, scaled by \( k \).
• If \( x \) decreases, \( y \) decreases in relation to the square of the decrease.

For example, if \( x \) is halved, as seen in the original problem, substituting \( \frac{x}{2} \) into the equation gives \( y_2 = k\left( \frac{x}{2} \right)^2 = \frac{y}{4} \). Thus, \( y \) would become one quarter of its original value.
Knowing how to set up and use such proportionality equations allows us to model and predict changes in different scientific and mathematical contexts.