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Chapter 3: Problem 88

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
In graphs of the form \(y=\frac{1}{x^{n}}\), an increase in even \(n\) values results in the graph flattening towards the x-axis, with this effect being most pronounced for x-values within the range of -1 to 1. The horizontal asymptote remains the x-axis for all these functions.

Step by step solution

01

Plot Initial Function

The first function to plot on the graph is \(y=\frac{1}{x^{2}}\). This sequence can be graphed using a graphing utility. The graph of \(y=\frac{1}{x^{2}}\) is a hyperbola opening upwards and downwards along the y-axis.
02

Plot Second Function

Next, plot the function \(y=\frac{1}{x^{4}}\) on the same graph, also a hyperbola but more flattened than the previous function.
03

Plot Third Function

Lastly, plot the function \(y=\frac{1}{x^{6}}\), it is again a hyperbola but is even more flattened than the second function. Take note of these observations.
04

Analyze the Graphs

Analyze the changes in the graphs. You will notice that as \(n\) increases, the graph gets more flattened towards the x-axis. The increase only affects the graph for x-values between -1 and 1, outside this range it has less impact. The horizontal asymptote, (i.e., the x-axis) remains the same for all.

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