Problem 9 Determine the asymptotes paralle... [FREE SOLUTION]

Chapter 20: Problem 9

Determine the asymptotes parallel to the $x$ - and $y$-axes for the function $$ x^{2} y^{2}=9\left(x^{2}+y^{2}\right) $$

Short Answer

Expert verified

Asymptotes are $ y = \pm 3 $ (horizontal) and $ x = \pm 3 $ (vertical).

Step by step solution

Rewrite the Equation

Starting with the given equation $ x^2 y^2 = 9(x^2 + y^2) $, we rewrite it to express it in a different form that could reveal potential asymptotes. Divide both sides of the equation by $ x^2 $ or $ y^2 $. We'll choose to divide by $ x^2 $, giving us $ y^2 = 9 + \frac{9y^2}{x^2} $. This shows how $ y^2 $ relates to $ x^2 $.

Consider Asymptotic Behavior as $ x \to \infty $

Assume $ x \to \infty $, such that the term $ \frac{9y^2}{x^2} \to 0 $. The equation simplifies to $ y^2 = 9 $, giving us $ y = 3 $ or $ y = -3 $. These are horizontal asymptotes parallel to the $ x $-axis.

Consider Asymptotic Behavior as $ y \to \infty $

Assume $ y \to \infty $, requiring us to adjust the equation by dividing through by $ y^2 $ this time. Divide both sides of the equation by $ y^2 $, yielding $ x^2 = 9 + \frac{9x^2}{y^2} $. As $ \frac{9x^2}{y^2} \to 0 $, we get $ x^2 = 9 $, resulting in $ x = 3 $ or $ x = -3 $. These are vertical asymptotes parallel to the $ y $-axis.

Review Asymptotes

Summarizing the results from the previous steps, we determine that the asymptotes of the function are $ y = 3 $, $ y = -3 $ (horizontal asymptotes), and $ x = 3 $, $ x = -3 $ (vertical asymptotes).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Asymptotes

Horizontal asymptotes occur when the behavior of a function as the variable moves towards infinity can be captured as a straight line parallel to the x-axis. In our exercise, the original equation is rewritten in terms of $ y^2 $, as $ y^2 = 9 + \frac{9y^2}{x^2} $. As $ x \to \infty $, the term $ \frac{9y^2}{x^2} $ becomes negligibly small, approaching zero.
This simplifies the equation to $ y^2 = 9 $.

This simplifies further to horizontal lines: $ y = 3 $ and $ y = -3 $.
These lines depict where the curves of the function become more linear as $ x $ becomes extremely large.

Understanding these lines is crucial because they help predict the function's behavior at extreme values of $ x $. By knowing $ y = \pm 3 $, students can better visualize the slope and curve direction as they stretch along the x-axis.

Vertical Asymptotes

Vertical asymptotes highlight the behavior of a function as it approaches large values of the other variable, $ y $. When the original function is rewritten, this time in terms of $ x^2 $, we see $ x^2 = 9 + \frac{9x^2}{y^2} $. Assuming $ y \to \infty $, the part $ \frac{9x^2}{y^2} $ limits to zero.
This changes the equation to $ x^2 = 9 $.

Consequently, we establish vertical lines: $ x = 3 $ and $ x = -3 $.
These lines represent the paths towards which the curves approach.

Vertical asymptotes are essential, as they indicate where a function might diverge, giving insight into the function's steepest regions. Knowing $ x = \pm 3 $ aids in comprehending how the graph behaves on reaching extreme y-values.

Infinite Behavior

Infinite behavior in functions refers to how a function acts as input values become larger and larger, either positively or negatively. The process involves observing the changes as one variable reaches infinity, thereby simplifying parts of the equation.

For horizontal asymptotes, as $ x \to \infty $, parts of the function become constant, depicting flatness or a level behavior, such as in $ y = 3 $ and $ y = -3 $.
Vertical asymptotes, depicted through $ x = 3 $ and $ x = -3 $, show where the function steepens intensely.

Infinite behavior exemplifies asymptotic lines, indicative of where the graph will tail off without ever crossing or touching indefinitely. Understanding this behavior is pivotal because it provides a clearer view of graph tendencies at extreme values, improving comprehension of function limits and the nature of their curves.

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91影视

Determine the asymptotes parallel to the \(x\) - and \(y\)-axes for the function $$ x^{2} y^{2}=9\left(x^{2}+y^{2}\right) $$

Short Answer

Step by step solution

Rewrite the Equation

Consider Asymptotic Behavior as \( x \to \infty \)

Consider Asymptotic Behavior as \( y \to \infty \)

Review Asymptotes

Key Concepts

Horizontal Asymptotes

Vertical Asymptotes

Infinite Behavior

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