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Chapter 2: Problem 123

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Short Answer

Expert verified
For even values of \(n\), increasing \(n\) causes the graph of \(y=\frac{1}{x^{n}}\) to approach 0 more rapidly as \(x\) moves away from zero. This results in a graph that is steeper around \(x=0\) and flatter as \(x\) moves away from zero.

Step by step solution

01

Graphing the Functions

Start by graphing the three functions \(y=\frac{1}{x^{2}}\), \(y=\frac{1}{x^{4}}\), and \(y=\frac{1}{x^{6}}\) on the same viewing rectangle. Carefully observe the shape, symmetry and characteristics of each graph.
02

Comparing the Graphs

Next, compare the graphs of \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}}\) and \(y=\frac{1}{x^{6}}\). Notice the differences and similarities between the functions, particularly in how fast they approach 0 as \(x\) increases or decreases.
03

Analyzing the Impact of the Exponent

Finally, analyze the impact of the exponent \(n\) on the graph of \(y=\frac{1}{x^{n}}\) for even values of \(n\). Note that as \(n\) increases, the function \(y=\frac{1}{x^{n}}\) approaches 0 more rapidly as \(x\) moves away from zero, either positively or negatively. This results in a graph that is steeper around \(x=0\) and flatter as \(x\) moves away from zero.

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