Problem 92 - A Graphical Approach to College Algebra

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

If \(x\) is halved, \(y\) becomes one fourth of its original value.

Step 1: Understanding the Relationship

Given that \(y\) is directly proportional to the second power of \(x\), we can express this relationship mathematically as \(y = kx^2\), where \(k\) is the constant of proportionality. This means \(y\) changes as \(x^2\) changes.

Step 2: Modify the Variable

We need to understand what happens when \(x\) is halved. If the original value of \(x\) is simply \(x\), then the new value will be \(\frac{x}{2}\).

Step 3: Substitute the New Value

Replace \(x\) in the equation \(y = kx^2\) with the halved value \(\frac{x}{2}\). This gives us: \[ y' = k\left(\frac{x}{2}\right)^2 \].

Step 4: Simplify the Expression

Simplify the expression \(y' = k\left(\frac{x}{2}\right)^2\) to find the new value of \(y\): \[ y' = k\frac{x^2}{4} = \frac{kx^2}{4} \].

Step 5: Compare Old and New Values

The original value of \(y\) was \(kx^2\). Now the new value of \(y\) is \(\frac{kx^2}{4}\). This shows that the new \(y\) is \(\frac{1}{4}\) of the original \(y\).

Key Concepts

Constant of Variation

In mathematics, when you come across terms such as "constant of variation," it refers to a specific fixed number in a direct variation equation. Let's dive into what this means in the context of direct proportionality. If a quantity depends directly on another, like in our problem where \( y \) is directly proportional to \( x^2 \), we write it mathematically as:

• \( y = kx^2 \)

Here, \( k \) is the constant of variation, also known as the constant of proportionality. It's crucial because it links the two variables, determining exactly how one variable increases or decreases as the other does. When you solve these types of problems, identifying this constant is a critical step. It's fixed for a particular relationship except when given specific conditions that might alter the basic setup, like changes to \( x \) or \( y \).
Understanding this concept helps simplify complex relationships and allows you to compare how changes to one variable impact another.

Quadratic Relationship

A quadratic relationship involves variables raised to the second power or squared. In our exercise, the relationship is showcased with \( y \) being directly proportional to \( x^2 \). This means that if the value of \( x \) changes, the value of \( y \) changes with the square of \( x \). Let's illustrate this with the exercise:

• Originally, the equation \( y = kx^2 \) describes how \( y \) changes based on \( x \).

Now, if \( x \) becomes \( \frac{x}{2} \), which means it is halved, the relation transforms to:

• \( y' = k\left(\frac{x}{2}\right)^2 \)

Breaking it down further, we compute:

• \( y' = k\frac{x^2}{4} = \frac{kx^2}{4} \)

Thus, when \( x \) is halved, \( y \) becomes a quarter of its original value, exhibiting how changes in \( x \) cause changes in \( y \) based on the square. This quadratic nature means that small changes in \( x \) can have magnified impacts on \( y \). Understanding quadratic relationships helps in recognizing how squared variables bring about different dynamics in mathematical problems.

Algebraic Expressions

Algebraic expressions allow us to represent real-world problems mathematically and make them easier to understand and solve. In this exercise, you're given an algebraic expression relating \( y \) and \( x \):

• \( y = kx^2 \)

This expression contains terms, coefficients, and variables combined through arithmetic operations. We use this to figure out how changes to \( x \) impact \( y \). The task involved modifying this expression by substituting the new value \( \frac{x}{2} \) for \( x \):

• \( y' = k\left(\frac{x}{2}\right)^2 \)

Then, simplifying it to understand the change:

• \( y' = \frac{kx^2}{4} \)

Simplifying algebraic expressions aids in revealing the inherent relationships between variables. They help in drawing clear conclusions about how one variable behaves relative to another. Effective handling of algebraic expressions is an essential skill for solving complex mathematical problems with logical clarity.