Problem 11 - 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: \(y = 125x^3\), where \(k = 125\), \(p = 3\).

Step 1: Identify the Function Components

First, understand that the given function is \(y = (5x)^3\). This is an expression where "\(5x\)" is raised to the power of 3.

Step 2: Expand the Expression

Expand the expression \((5x)^3\) to \(5^3 \cdot x^3 = 125x^3\). This represents the expression where both the coefficient and the variable are raised to the power indicated.

Step 3: Confirm Power Function Form

Identify that the expanded form \(y = 125x^3\) matches the power function form \(y = kx^p\), where \(k = 125\) and \(p = 3\).

Step 4: State the Power Function Form

The power function form is \(y = 125x^3\), with \(k = 125\) and \(p = 3\). This confirmation shows it is indeed a power function.

Key Concepts

Function Components

To determine what makes up the components of a function, start by looking at each part of the expression provided. For \(y = (5x)^3\): - **Base**: The base of our power is \(5x\), meaning the combination of the coefficient (5) with the variable (x). This is the central part around which the expression rotates.- **Exponent**: The exponent in this function is 3, indicating that the base \((5x)\) is multiplied by itself three times.Identifying these components helps you understand the building blocks of the function, how they interact, and lay a foundation for expanding and transforming the expression.

Expand Expression

Expanding an expression involves removing the parentheses and writing it out fully to reveal hidden components. Let's expand \((5x)^3\). When you expand, you rewrite it as:- \((5x) \times (5x) \times (5x)\). This can be further broken down using the properties of exponents:- Which implies \(5^3\) multiplied by \(x^3\).- By computing, we get \(5 \times 5 \times 5 = 125\), hence - \(125x^3\).The expanded form \(125x^3\) shows how each component, the coefficient, and the variable, reacts to the power they are raised to, fully revealing the function's structure.

Coefficients and Exponents

Coefficients and exponents are key players in any power function. The coefficient is the number that precedes the variable, affecting the function's steepness or growth rate. Here, the coefficient is - **125**, derived from expanding the base of \((5x)^3\). The exponent, on the other hand, indicates the degree of the power to which the variable is raised. In this function:- The exponent is - **3**, which signifies how many times the base is used as a factor. This relationship between coefficients and exponents determines the behavior of the function: higher coefficients stretch or compress the graph, while larger exponents affect the curvature and directionality.

Power Function Form

Identifying a power function involves writing the function expression in a specific format. A power function can typically be recognized by the form \(y = kx^p\), where both \(k\) and \(p\) are constants. In the example given, the expanded function is - \(y = 125x^3\), which matches the power function form. Here, we identify:- **\(k = 125\)**: The constant multiplier of the variable base, signifying the strength or magnitude of the power's effect.- **\(p = 3\)**: The exponent representing how many times the base variable is a factor of the function.Recognizing these parts confirms that the function is indeed a power function, providing insight into both the algebraic structure and possible graphical behavior of the function described. This concrete form gives a straightforward way to analyze and predict behavior in broader mathematical problems.