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Chapter 3: Problem 87

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 \(y=\frac{1}{x^{n}} ?\)

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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