Vertical and Horizontal Asymptotes

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Discussion Overview

The discussion revolves around the concept of vertical and horizontal asymptotes in the context of an inverse variation equation, specifically examining the equation y = (1 / (x - 3)) - 6. Participants are exploring how to identify these asymptotes and the reasoning behind their values.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant seeks clarification on the asymptotes of the equation y = (1 / (x - 3)) - 6, expressing confusion about the values y = -6 and x = 3.
  • Another participant prompts the first to consider the behavior of a fraction as the denominator varies, suggesting that understanding this behavior is key to identifying asymptotes.
  • A subsequent reply reiterates the idea that the fraction decreases as the denominator increases and increases as the denominator approaches zero, but seeks more specificity in the explanation.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as participants are still exploring the concepts and reasoning behind asymptotes without definitive conclusions.

Contextual Notes

Participants have not yet established a clear understanding of the definitions and implications of vertical and horizontal asymptotes, and there are unresolved questions about the reasoning behind the identified values.

mathewslauren
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In an inverse variation equation, what are the asymptotes and how do you find them? For example,
I was given the equation: y= [1 \ (x - 3)] - 6 and asked to find the vertical and horizontal asymptote.
I don't really understand what they are and why y= -6 and x=3. Thanks for any help!
 
Last edited:
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Hello and welcome to MHB, mathewslauren! (Wave)

We are given:

$$y=\frac{1}{x-3}-6$$

Now, before we discuss asymptotes, think about if you have a fraction, and you hold the numerator constant, and let the denominator vary. What happens to the value of the fraction if the denominator get larger and larger, without bound...where is the value of the fraction itself headed...and likewise, what if we let the denominator get closer and closer to zero...what happens to the value of the fraction then?
 
The fraction would get smaller as the denominator increases, and larger as it decreases.
 
mathewslauren said:
The fraction would get smaller as the denominator increases, and larger as it decreases.

Well, that's true, but can you be more specific?
 

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