Problem 10 - Applied Calculus

Question

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=(5 x)^{3}$$

Step-by-Step Solution

Answer

Yes, it's a power function with \( k = 125 \) and \( p = 3 \).

Step 1: Recognize the Form of the Given Function

The given function is \( y = (5x)^3 \). Observe that it involves the power operation on a product. Our goal is to determine if this fits the general power function form \( y = kx^p \).

Step 2: Simplify the Expression

To test if \( y = (5x)^3 \) can be rewritten in the form \( y = kx^p \), we'll expand \( (5x)^3 \). Applying the property of exponents, \((ab)^m = a^m b^m\), we get: \[ (5x)^3 = 5^3 \, x^3 \].

Step 3: Compute the Constant 'k'

Now calculate \( 5^3 \) which equals 125. Thus, the expression simplifies to \( y = 125x^3 \).

Step 4: Identify 'k' and 'p' for the Power Function

The function \( y = 125x^3 \) is now clearly in the form \( y = kx^p \). By comparing, we identify \( k = 125 \) and \( p = 3 \).

Step 5: Conclude the Power Function Recognition

Since the function is of the form \( y = kx^p \), it is indeed a power function with \( k = 125 \) and \( p = 3 \).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when dealing with power functions. Exponents are a mathematical tool used to denote repeated multiplication of a base number. In the given function, we began with \( y = (5x)^3 \). Here, the exponent 3 applies to both the constant 5 and the variable \( x \), indicating that both are multiplied by themselves three times. This results in \( (5x)^3 \) being expanded to \( 5^3 \cdot x^3 \).

• One important exponent rule is \((ab)^m = a^m \cdot b^m\), which was applied here to distribute the exponent to both components within the parentheses.
• This rule helps simplify expressions by separating the constant and variable parts before evaluating their powers individually.

Another key rule states that when multiplying powers with the same base, you add exponents. Although this specific rule wasn't used directly in this exercise, it’s a foundational component of working with exponents. Understanding these rules allows smooth transitions between different expressions and eases the calculation process when simplifying power functions.

Function Simplification

Function simplification involves rewriting an expression in a simpler or more standard form. In this exercise, the task was to determine if the given function \( y = (5x)^3 \) is a power function. To do this, we used properties of exponents to simplify the function.First, we expanded the expression using the rule: \( (ab)^m = a^m b^m \). By doing this, \( (5x)^3 \) was rewritten as \( 5^3 x^3 \). This separation allows us to clearly see that \( y = 125x^3 \) matches the form of a power function, \( y = kx^p \).

• Calculating \( 5^3 \) gives us 125, which becomes the constant \( k \) in the power function format.
• The variable \( x \) is raised to the third power, identifying \( p = 3 \).

By simplifying the function, we make it easier to identify the constants and powers required to fit it into the standard power function form. This simplification is often needed for comparison, further calculations, or when solving equations.

Identifying Constant and Exponent

Identifying constants and exponents is vital for categorizing functions and understanding their characteristics. A power function is one of the simplest categories of functions and has the form \( y = k x^p \), where \( k \) and \( p \) are constants. In our exercise, after simplifying \( y = (5x)^3 \) to \( y = 125x^3 \), we could easily determine the values of the constants. Here, \( k \) is the coefficient of \( x^p \), which we found to be 125, and \( p \) is the exponent on the variable, which is 3.

• The constant \( k \) often represents a proportionality factor in the relationship between \( y \) and \( x \).
• The exponent \( p \) indicates the degree or the nature of the relationship's dependence on \( x \), defining the function's curve.

Clear identification of these constants allows you to fully describe the behavior of the power function. Knowing \( k \) and \( p \) tells you how changes in \( x \) affect the output value \( y \), which is critical in mathematical modeling and analysis. Always break down a function to easily spot these values to help in graphing or further calculations.