Problem 88 - College Algebra

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

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.