Problem 8 - 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 \(p\). $$y=\frac{2 x^{2}}{10}$$

Step-by-Step Solution

Answer

The function is a power function: \(k = 0.2\), \(p = 2\).

Step 1: Simplify the expression

The given function is \(y = \frac{2x^2}{10}\). First, simplify the coefficient by dividing the numerator by the denominator. \(\frac{2}{10} = 0.2\), so the function can be rewritten as \(y = 0.2x^2\).

Step 2: Identify power function form

A power function can be written in the form \(y = kx^p\). In the simplified expression \(y = 0.2x^2\), compare this with the power function form. Here we identify that \(k = 0.2\) and \(p = 2\).

Step 3: Conclusion: Determine power function status

Since the expression \(y = 0.2x^2\) matches the form of \(y = kx^p\), the given function is a power function with \(k = 0.2\) and \(p = 2\).

Key Concepts

Function Simplification

Understanding function simplification is crucial for making complex problems more manageable. In the given exercise, our task was to simplify the function \(y = \frac{2x^2}{10}\). Simplifying means expressing a mathematical idea in a simpler form without losing its original meaning.
To simplify the function, we start by addressing the fraction. The numerator here is \(2x^2\) and the denominator is \(10\). By dividing the constant in the numerator by the constant in the denominator, we simplify the coefficient. Here, performing \(\frac{2}{10}\) gives us \(0.2\). This makes our function \(y = 0.2x^2\).
The goal of simplification is to make the function easier to analyze and work with, especially when identifying forms like the power function form. Remember, although the expression looks different, it maintains the same value and properties as the original form.

Power Function Form

A power function is one of the fundamental expressions in mathematics. It's written in the form \(y = kx^p\). This means "\(y\) equals \(k\) times \(x\) raised to the power of \(p\)."
In the power function form:

• \(k\) is a constant coefficient.
• \(p\) is the exponent that determines the degree of the function.
• \(x\) is the variable.

By identifying these components in a given function, we can determine whether it adheres to the power function structure. In our simplified function \(y = 0.2x^2\), we see this clear alignment.
Here, \(k = 0.2\) and \(p = 2\). Notice that any function of this form will correspond to a power function, which can help us better understand and predict its behavior when graphed or used in calculations.

Constant Coefficient

The term constant coefficient, represented by \(k\) in the power function form \(y = kx^p\), adds a fixed value that scales the function. This coefficient is not influenced by the variable \(x\); rather, it is a constant factor throughout the entire function.
In our specific example, \(y = 0.2x^2\), \(k\) is \(0.2\). This means that no matter what value \(x\) takes, the results will always be scaled by this coefficient.
Understanding the role of the constant coefficient is important because it influences the height or amplitude of the function on a graph. A larger \(k\) implies a steeper graph, while a smaller \(k\) leads to a flatter graph. Thus, by changing \(k\), you adjust how tightly or loosely the function hugs its baseline—without altering its underlying shape.