Problem 45 - Intermediate Algebra

Question

Suppose that \(y\) varies directly as the square of \(x\). Does doubling the value of \(x\) also double the value of \(y\) ? Explain your answer.

Step-by-Step Solution

Answer

No, doubling \( x \) quadruples \( y \), not doubles it.

Step 1: Define Direct Variation

To solve the problem, first recognize that direct variation as a square means the relationship between two variables can be expressed as \( y = kx^2 \), where \( k \) is a constant.

Step 2: Analyze the Effect of Doubling x

Now, suppose the value of \( x \) is doubled. The new value of \( x \) becomes \( 2x \). Substitute \( 2x \) into the equation \( y = kx^2 \), giving us \( y = k(2x)^2 \).

Step 3: Simplify the Expression

Simplify \( y = k(2x)^2 \): \[ y = k(4x^2) = 4(kx^2) \] This shows that when \( x \) is doubled, \( y \) is multiplied by 4.

Step 4: Conclude the Relationship

The expression \( y = 4(kx^2) \) shows that \( y \) becomes four times its original value, not double. Therefore, doubling \( x \) does not double \( y \); it quadruples it.

Key Concepts

Relationship Between Variables

In mathematics, direct variation describes a specific type of relationship between two variables where one variable is directly proportional to a function of another variable. This means that the value of one variable changes in direct relation to the other. For example, when variable \( x \) increases, variable \( y \) will also increase, provided they are directly related. In our exercise, we are given that \( y \) varies directly with the square of \( x \), which is expressed by the formula \( y = kx^2 \). Here, \( k \) is a constant that remains the same regardless of the values of \( x \) or \( y \). This highlights how changes in \( x \) influence changes in \( y \) consistently, following the equation.

Quadratic Relationship

A quadratic relationship is a type of polynomial relationship, specifically one where the highest exponent of the variable is 2. This is different from linear relationships (with their straight line graphs), as a quadratic relationship will form a parabolic curve when graphed. In the equation \( y = kx^2 \), the variable \( x \) is squared, representing a quadratic relationship between \( x \) and \( y \). If we double the value of \( x \) and substitute it into the equation, \( y = k(2x)^2 = k(4x^2) \). The simplification shows that \( y \) changes in a way that corresponds to the square of the scaling factor of \( x \), leading \( y \) to become four times larger rather than simply doubling. This illustrates the significant impact a quadratic relationship can have on the values of the involved variables.

Proportionality Constant

In the equation \( y = kx^2 \), the term \( k \) is known as the proportionality constant. It serves a crucial role in the relationship between \( x \) and \( y \). This constant determines the rate at which \( y \) changes as \( x \) changes. It effectively scales the value of \( x^2 \) to match the value of \( y \).Regardless of the value of \( x \), \( k \) remains unchanged, signifying that it provides a direct measure of how \( x \) and \( y \) are proportionally related. For instance, in our scenario, knowing \( k \) allows us to predict how \( y \) will change if \( x \) is altered. This constancy of \( k \) is what makes calculating potential changes in \( y \) straightforward when modifications to \( x \) occur, such as doubling \( x \), to determine that \( y \) becomes four times its original value.