Problem 87 - College Algebra

Question

Use a graphing utility to graph \(y=\frac{1}{x}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Step-by-Step Solution

Answer

By increasing the odd power 'n' in the function \(y=\frac{1}{x^{n}}\), the steepness of the curve increases near x=0, while the rest of the plot flattens out more. All of these graphs show origin symmetry, typical for functions with odd exponents.

Step 1: Graphing the functions

First, start by plotting the functions \(y=\frac{1}{x}, y=\frac{1}{x^{3}}, y=\frac{1}{x^{5}}\). This can be done by inputting the functions into a graphing utility.

Step 2: Comparing the Plots

Compare the three plots on the graph. Key characteristics to observe include how each function behaves near x=0, and changes in the curve's slope as x values increase or decrease.

Step 3: Analyzing the impact of changing 'n'

Observe how the graphs vary when the exponent n in \(y=\frac{1}{x^{n}}\) changes. Particularly, we pay attention to how changes in 'n' affect the sharpness of the curve near x = 0, and how this changes the symmetry of the plot. With odd exponents, the graphs possess symmetry about the origin.