Problem 13 - Algebra Form and Function

Question

In Exercises \(13-16,\) what happens to \(y\) when \(x\) is doubled? Here \(k\) is a positive constant. $$ y=k x^{3} $$

Step-by-Step Solution

Answer

Answer: When x is doubled, the value of y becomes 8 times its original value.

Step 1: Double the value of x

First, we replace x with 2x in the given equation: $$ y = k(2x)^3 $$

Step 2: Simplify the equation

Now, simplify the equation to observe the change in y: $$ y = k(2^3x^3) = k(8x^3) $$

Step 3: Describe the change in y

When x is doubled, we can see that y becomes 8 times its original value because of the factor 8 in the simplified equation: $$ y = 8kx^3 $$ So, when x is doubled, y is multiplied by 8.

Key Concepts

Direct Variation

Direct variation is a fundamental concept that describes a relationship where one variable changes directly with another variable. In simpler terms, if one variable increases, the other increases by a constant proportion, and if it decreases, the same happens. This concept is expressed mathematically as:

• \( y = kx \)

where \( y \) and \( x \) are the variables in question, and \( k \) is their constant of proportionality. In the context of the problem, the equation \( y = kx^3 \) is a direct variation because \( y \) changes as a cubic function of \( x \).
When \( x \) is doubled in the equation, the change in \( y \) is still proportional to the change in \( x^3 \), which highlights direct variation's nature. In this particular exercise, the fact that doubling \( x \) results in \( y \) becoming 8 times its original value is indicative of this consistent proportionality.

Constant of Proportionality

The constant of proportionality, often represented as \( k \), plays a crucial role in direct variation. It determines the exact scaling relationship between \( x \) and \( y \). In our problem, \( k \) is what links \( x^3 \) to \( y \), creating a consistent multiplier effect.
When \( x \) is altered, \( k \) remains constant, ensuring the relationship between \( x \) and \( y \) does not change in its nature, merely its magnitude.
Think of \( k \) as a scaling factor or the route on how changes in \( x \) directly affect \( y \). Even though \( x^3 \) involves an exponent, \( k \) remains steady, allowing us to predictably determine \( y \) when \( x \) is modified. In our example, twice the value of \( x \) means \( y \) will become eight times bigger because of how exponents influence the constant multiplier.

Exponents

Exponents are mathematical notations that indicate how many times a number, known as the base, is multiplied by itself. They are crucial in expressing polynomial equations like \( y = kx^3 \). Here, the exponent 3 means \( x \) is multiplied by itself three times to get \( x^3 \).
The effect of exponents in equations is significant because small changes in the base \( x \) can lead to substantial changes in \( y \). For example, when \( x \) is doubled, \( (2x)^3 = 8x^3 \) because \( 2^3 = 8 \). Thus, doubling \( x \) results in an eightfold increase in \( y \).
Understanding exponents helps in predicting how outputs in equations will behave when inputs are adjusted. It simplifies complex multiplication and allows for easier computation of variables changing by powers, reinforcing the robust nature of quadratic and higher-order polynomial equations.