Vertical and Horizontal Asymptotes

In summary, the conversation discusses the concept of asymptotes in an inverse variation equation. Asymptotes are values that the equation approaches but never reaches. In this case, the vertical asymptote is x=3 and the horizontal asymptote is y=-6. This information can be found by considering the behavior of the fraction as the denominator gets larger and smaller.
  • #1
mathewslauren
3
0
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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  • #2
Hello and welcome to MHB, mathewslauren! (Wave)

We are given:

\(\displaystyle y=\frac{1}{x-3}-6\)

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?
 
  • #3
The fraction would get smaller as the denominator increases, and larger as it decreases.
 
  • #4
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?
 

FAQ: Vertical and Horizontal Asymptotes

What are Vertical and Horizontal Asymptotes?

Vertical and Horizontal Asymptotes are imaginary lines that a graph approaches but never touches. They are used to describe the behavior of a function as the input values approach certain values.

How do you find the Vertical Asymptotes of a function?

To find the Vertical Asymptotes of a function, set the denominator of the function equal to zero and solve for the input values that make the denominator equal to zero. These input values will be the Vertical Asymptotes.

How do you find the Horizontal Asymptotes of a function?

To find the Horizontal Asymptotes of a function, you need to look at the degrees of the numerator and denominator of the function. If the degree of the numerator is less than the degree of the denominator, the Horizontal Asymptote is y = 0. If the degrees are equal, the Horizontal Asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no Horizontal Asymptote.

What is the difference between a Vertical and Horizontal Asymptote?

The main difference between a Vertical and Horizontal Asymptote is the direction in which the graph approaches the line. A Vertical Asymptote is a vertical line that the graph approaches from either the left or right side, while a Horizontal Asymptote is a horizontal line that the graph approaches from either the top or bottom.

Why are Vertical and Horizontal Asymptotes important in graphing functions?

Vertical and Horizontal Asymptotes provide important information about the behavior of a function. They help us understand the overall shape of the graph and identify any potential discontinuities or undefined points. They also help us make predictions about the behavior of the function as the input values approach certain values.

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