Problem 122 Use a graphing utility to graph ... [FREE SOLUTION]

91影视

Precalculus Essentials

Robert Blitzer

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 2: Problem 122

Use a graphing utility to graph $y=\frac{1}{x}, y=\frac{1}{x^{3}},$ and $\frac{1}{x^{5}}$ in the same viewing rectangle. For odd values of $n,$ how does changing $n$ affect the graph of $y=\frac{1}{x^{n}} ?$

Short Answer

Expert verified

For functions of the form $y=\frac{1}{x^n}$ where $n$ is an odd number, as $n$ increases, the graph of the function becomes flatter between the quadrants and sharply increases as it gets closer to $x = 0$. The function will cover all four quadrants.

Step by step solution

Graphing $y=\frac{1}{x}$

Using a graphing utility, graph the function $y=\frac{1}{x}$. Notice the characteristics of the graph. It should have a hyperbolic shape, with two distinct parts on the 1st and 3rd quadrant, getting closer and closer to the axes but never reaching them. This goes to infinity when $x$ approaches zero and goes to zero when $x$ approaches infinity or negative infinity.

Graphing $y=\frac{1}{x^3}$

Now, using the same utility, graph the function $y=\frac{1}{x^3}$. This graph should also have a similar hyperbolic shape. However, compared to the previous one, this graph will penetrate the 2nd and 4th quadrant. This graph goes to negative infinity when $x$ is a large negative value, positive infinity when $x$ is a large positive value, and also goes to infinity when $x$ approaches zero.

Graphing $y=\frac{1}{x^5}$

Lastly, graph $y=\frac{1}{x^5}$. The graph of this equation should look fairly similar to that of the $y=\frac{1}{x^3}$ graph. It also will have a hyperbolic shape and penetrate all four quadrants.

Comparing graphs

After plotting the three functions, when $n$ is odd, the graph of $y = \frac{1}{x^n}$ will always cross through all four quadrants. The higher the odd value of $n$, the closer the graph will be to the axes between the quadrants. This means that the graph gets flatter in the intervals (-鈭�, 0) and (0, 鈭�), but sharply increases as it approaches $x = 0$, reaching positive or negative infinity. This is because raising a positive number to an odd power results in a positive number, and raising a negative number to an odd power results in a negative number.

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91影视

Use a graphing utility to graph \(y=\frac{1}{x}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Short Answer

Step by step solution

Graphing \(y=\frac{1}{x}\)

Graphing \(y=\frac{1}{x^3}\)

Graphing \(y=\frac{1}{x^5}\)

Comparing graphs

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