Problem 113 - College Algebra Essentials

Question

a. Graph the functions \(f(x)=x^{n}\) for \(n=2,4,\) and 6 in a \([-2,2,1]\) by \([-1,3,1]\) viewing rectangle. b. Graph the functions \(f(x)=x^{n}\) for \(n=1,3,\) and 5 in a \([-2,2,1]\) by \([-2,2,1]\) viewing rectangle. c. If \(n\) is positive and even, where is the graph of \(f(x)=x^{n}\) increasing and where is it decreasing? d. If \(n\) is positive and odd, what can you conclude about the graph of \(f(x)=x^{n}\) in terms of increasing or decreasing behavior? e. Graph all six functions in a \([-1,3,1]\) by \([-1,3,1]\) viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?

Step-by-Step Solution

Answer

a. The graphs of \(f(x)=x^{2}\), \(f(x)=x^{4}\) and \(f(x)=x^{6}\) within the viewing rectangle [-2, 2, 1] by [-1, 3, 1] are U-shaped. b. the graphs of \(f(x)=x^{1}\), \(f(x)=x^{3}\) and \(f(x)=x^{5}\) within the viewing rectangle [-2, 2, 1] by [-2, 2, 1] are increasing from left to right. c. For a positive and even \(n\), \(f(x)=x^{n}\) is decreasing for \(x < 0\), and increasing for \(x > 0\). d. For a positive and odd \(n\), \(f(x)=x^{n}\) is always increasing. e. The graph becomes steeper as the value of n increases. For negative x-values, odd power functions decrease steeply, whereas even power functions flatten.

Step 1: Graphing even functions

Plot the functions \(f(x) = x^{2}\), \(f(x) = x^{4}\) and \(f(x) = x^{6}\) within the viewing rectangle [-2, 2, 1] by [-1, 3, 1].

Step 2: Graphing odd functions

Plot the functions \(f(x) = x^{1}\), \(f(x) = x^{3}\), and \(f(x) = x^{5}\) within the viewing rectangle [-2, 2, 1] by [-2, 2, 1].

Step 3: Analyzing increasing and decreasing behavior for even functions

For a positive and even \(n\), \(f(x) = x^{n}\) is decreasing for \(x < 0\), and increasing for \(x > 0\). At \(x = 0\), we have minimum points, which can be observed in the graphs of step 1 as well.

Step 4: Analyzing increasing and decreasing behavior for odd functions

For a positive and odd \(n\), \(f(x) = x^{n}\) is always increasing. This can be observed in the graphs of step 2.

Step 5: Graphing all six functions

Plot all six functions \(f(x) = x^{n}\), where n is 1, 2, 3, 4, 5, and 6 within the viewing rectangle [-1, 3, 1] by [-1, 3, 1] as in step 1 and step 2.

Step 6: Analyzing the graphs

From the graphs in step 5, it can be seen that as the value of n increases, the graph becomes steeper. For negative x-values, the odd power functions tend to decrease steeply, whereas even power functions tend to flatten.