Problem 119 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graphs of \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{n}}\) reveal that as \(n\) increases, the function flattens around \(x=0\) and the steepness of the descent toward the x-axis from both positive and negative directions increases. Thus, for even values of \(n\), increasing \(n\) squeezes the function closer to the x-axis, but also makes the graph steeper on either side of the y-axis.

Step 1: Graph the Functions

Use any graphing utility to plot \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\). Ensure that these graphs are drawn on the same scale/viewing rectangle, in order to facilitate the comparison of their shapes.

Step 2: Analyze the Graphs

Now carefully study these graphs to understand how they vary with the increase in the exponent \(n\). Pay attention to the overall shape, the steepness of the curves and any points of inflection.

Step 3: Conclusion

Conclude by summarizing the observed changes as the exponent \(n\) in the function \(y=\frac{1}{x^{n}}\) increases. Note the pattern that emerges with even exponents.