Problem 34 - Algebra for College Students

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

Doubling \( x \) quadruples, not doubles, \( y \), because \( y \) varies with the square of \( x \).

Step 1: Understand Direct Variation

In a direct variation relationship, as one variable increases, the other one increases by a consistent factor. Here, it is given that \( y \) varies directly with the square of \( x \). This means we can express this relationship with the equation \( y = kx^2 \), where \( k \) is the constant of variation.

Step 2: Analyze the Effect of Doubling x

Suppose we double the value of \( x \), so the new value is \( 2x \). We substitute this into the equation: \( y = k(2x)^2 \). Calculating this gives us \( y = k \times 4x^2 \).

Step 3: Compare the New Value of y

In the original equation, \( y = kx^2 \). After doubling \( x \), we have \( y = 4kx^2 \). This shows that the new value of \( y \) is four times the original value, not double.

Key Concepts

Constant of Variation

In mathematics, when we say that one variable varies directly as another, we’re talking about a direct proportional relationship. The "constant of variation," often denoted by the letter \( k \), plays a critical role in this relationship. It provides the consistent factor by which one variable changes in response to another.
For a direct variation, we express the relationship between the variables as \( y = kx^2 \) if \( y \) varies directly with the square of \( x \). Here, \( k \) acts as the constant that scales the change.
When solving problems related to direct variation, finding \( k \) is essential if you know specific points, or if you wish to understand the scaling effect precisely. Remember, \( k \) remains unchanged as long as the relationship between the variables is direct and without external alterations. This stability allows the relationship to be predictable across different values of \( x \).

Square of a Variable

The square of a variable, like \( x^2 \), means you multiply the variable by itself. This operation can significantly change how a variable behaves in functional relationships.
For example, when \( y \) is proportional to \( x^2 \), the relationship becomes more non-linear. If you increase \( x \), the effect on \( y \) will be amplified due to the squaring.
Understanding what squaring does helps clarify why doubling \( x \) does not just double \( y \). Instead, it quadruples \( y \) in the equation \( y = kx^2 \), because doubling \( x \) becomes \( (2x)^2 = 4x^2 \). Whenever you square a variable, especially in direct variations, expect accelerated changes in response to any scaling of \( x \).

Effect of Scaling a Variable

Scaling a variable means multiplying it by a factor. In direct variation scenarios, particularly when dealing with squares, the impact of scaling is exponential rather than linear.
Let's say we scale \( x \) by a factor of 2. First, we substitute \( 2x \) in place of \( x \) in the equation \( y = kx^2 \). The adjusted expression becomes \( y = k(2x)^2 = 4kx^2 \), showing how \( y \) becomes four times larger, illustrating the squared factor's amplification.
To understand this better:

• If you scale \( x \) by any factor \( a \), the effect on \( y \) will be \( a^2 \) times.
• This means if \( a = 3 \), \( y \) increases by 9 times because \( 3^2 = 9 \).

Scaling directly impacts the responsiveness of the equation, showing how even simple scaling can lead to large overall differences in output.