Problem 12 - 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's a power function: \(y = \frac{1}{5}x^1\) with \(k=\frac{1}{5}\) and \(p=1\).

Step 1: Understand the Power Function

A power function is traditionally expressed in the form \(y = kx^p\), where \(k\) and \(p\) are constants, \(x\) is the variable, and \(k eq 0\). The goal is to express the given function in this form and identify \(k\) and \(p\).

Step 2: Convert the Given Function

The given function is \(y = \frac{x}{5}\). This expression needs to be converted to the standard form of a power function. It can be rewritten as: \(y = \frac{1}{5}x\).

Step 3: Express in Power Function Form

In this form \(y = \frac{1}{5}x\), it can be seen that it fits the power function form \(y = kx^p\) with \(k = \frac{1}{5}\) and \(p = 1\), since \(x\) is equivalent to \(x^1\).

Step 4: State Values of k and p

The values extracted from the expression \(y = \frac{1}{5}x\) are \(k = \frac{1}{5}\) and \(p = 1\). Thus, \(y = \frac{1}{5}x^1\) is indeed a power function.

Key Concepts

Mathematical Constants

Mathematical constants are values that remain the same throughout equations and calculations. They are essential components in various mathematical functions and can help solve numerous mathematical problems. In power functions, constants are fundamental for defining the form and behavior of the function.

When we express a power function in the form \(y = kx^p\), \(k\) and \(p\) are constants:

• \(k\) is the coefficient, a constant that stretches or compresses the graph of the function. If \(k\) is positive, the graph will ascend; if negative, it will descend.
• \(p\) is the exponent, representing the power to which the variable \(x\) is raised. The exponent determines the shape and direction of the graph.

In the given problem, for instance, the function \(y = \frac{1}{5}x\) can be rewritten as \(y = kx^p\) with \(k = \frac{1}{5}\) and \(p = 1\). Here, the constant \(k\) indicates the graph's slope, while \(p\) implies a linear relationship, as the exponent is 1.Understanding constants and their roles is crucial in analyzing and applying power functions effectively.

Function Transformation

Function transformation involves altering a function's equation to change its graph's position, size, or orientation according to specific rules. In power functions specifically, transformation helps in adjusting functions into a standard recognizable form like \(y = kx^p\).

Transformations can include:

• Vertical shifts, moving the graph up or down by adding or subtracting a constant.
• Horizontal shifts, moving the graph left or right by adding or subtracting a constant within the exponent.
• Scaling, which involves stretching or compressing the graph either vertically (through \(k\) values) or horizontally (via exponent changes).
• Reflections, which occur by changing the sign of \(k\), flipping the graph across the x-axis.

In the original problem, the function \(y = \frac{x}{5}\) was transformed by recognizing it as \(y = \frac{1}{5}x\) without needing any additional transformations beyond adjusting into a standard power function format.
By mastering function transformations, students can better understand and manipulate functions, allowing for easier graphing and interpretation of results.

Exponents

Exponents are mathematical notations indicating how many times a base number is multiplied by itself. In power functions, exponents play a critical role, defining the degree and directionality of the curve represented by the function.

Here's a quick rundown on understanding exponents in this context:

• An exponent of 1 suggests a linear function, as seen in this exercise with \(y = \frac{1}{5}x^1\).
• Positive exponents indicate growth; as \(x\) increases, \(y\) increases, showing an upward curve.
• Negative exponents invert the relationship, leading to functions that decrease as \(x\) increases.
• Fractional exponents imply roots, such as \(x^{1/2}\) representing \(\sqrt{x}\).

In our exercise, recognizing that the exponent \(p = 1\) directly influences how we interpret the function as a simple line shows the power of exponents in defining mathematical expressions. Proper handling of exponents helps in clearly identifying the nature of the function, whether it's growth, decay, linear, or another form.
By grasping the concept of exponents, you can solve complex problems and understand a wide range of mathematical functions effectively.