Problem 1 - 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 \(p\) $$y=\frac{x}{5}$$

Step-by-Step Solution

Answer

Yes, it is a power function with \(k=\frac{1}{5}\) and \(p=1\).

Step 1: Identify the given function

The given function is \( y = \frac{x}{5} \). We need to determine if this can be expressed as a power function of the form \( y = kx^p \).

Step 2: Simplify the function to check for power function form

Rewrite the given function \( y = \frac{x}{5} \) in a way that fits the form \( y = kx^p \). Specifically, we can write it as \( y = \frac{1}{5}x^1 \), which resembles the form of a power function where \( k = \frac{1}{5} \) and \( p = 1 \).

Step 3: Verify the power function

Verify that the rewritten function \( y = \frac{1}{5}x^1 \) indeed represents a power function. In this case, \( k \) is a constant \( \frac{1}{5} \) and \( p \) is the exponent \( 1 \), confirming it is a power function.

Key Concepts

Power Function Form

A power function has the form \( y = kx^p \), making it a crucial component in understanding different types of mathematical equations and their behavior. When we say something is a power function, we're referring to any function where a single term is raised to a constant exponent.

• Form Elements: It must have one term in the form of a variable raised to the power of an exponent.
• Variable: Typically, the variable involved is \( x \), but it could be any other variable.
• Simplicity: Unlike other polynomials that may have a sum of multiple terms, a power function only focuses on one term with an exponent.

When given a function, the task is to see if it fits this format. For example, transforming \( y = \frac{x}{5} \) into \( y = \frac{1}{5}x^1 \) proves it can be rewritten in power function form. This step helps in analyzing and classifying different functions and also indicates how they might behave or be manipulated mathematically.

Constant Coefficient

In the context of power functions, the constant coefficient \( k \) plays a vital role in scaling the effect of the function. It is the numerical part that stays constant regardless of the value of the variable \( x \).

• Value of \( k \): It's the number that multiplies the variable raised to a power. For example, in \( y = \frac{1}{5}x^1 \), \( k = \frac{1}{5} \).
• Influence: The coefficient determines the function's vertical stretch or shrink. It impacts the steepness of the graph, affecting how quickly the output increases or decreases.

Understanding \( k \) helps predict how the function behaves as \( x \) changes. If \( k \) is larger than 1, the function's growth multiplies rapidly; if between 0 and 1, it's less aggressive.

Exponent

The exponent \( p \) in a power function determines the degree and direction of the function. This impacts the curvature and overall shape of the graph.

• Role of \( p \): In \( y = kx^p \), \( p \) signifies the power to which the variable is raised. In \( y = \frac{1}{5}x^1 \), \( p = 1 \).
• Impact on Graph: A positive exponent leads to a standard power function shape, whereas a negative \( p \) would flip the curve, introducing new patterns like reciprocal functions.
• Nature of p:**: Higher exponents mean curves will rise or drop faster as \( x \) moves away from zero. Linear: For \( p=1 \), the function is linear. The graph is a straight line.

Exploring the exponent allows you to comprehend how rapidly the outputs may change, adding depth to your understanding of function behavior in different mathematical contexts.