Asymptotes curve (1-2x)/(3x+5)

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In summary, when finding the vertical asymptote, we should check the denominator and see which point doesn't exist. And when finding the horizontal asymptote, we should take the limit of the function as x tends to infinity and see the relation between the limit value and the horizontal asymptote. Additionally, for rational functions of polynomials, we can divide both the numerator and denominator by the highest power of x and take the limit as x approaches infinity to find the horizontal asymptote. It is also important to note that an asymptote is a line or y-value that the graph approaches, and should be written as y=k rather than just k.
  • #1
JohnnyPhysics
I the curve (1-2x)/(3x+5). I have been asked to find the verticle and horizontal asymptotes. Can anyone help me with a strategy?
 
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  • #2
Here are some hints.

Vertical asymptote: check the demominator, see which point doesn't exist.

horizontal asymptote: take limit of f(x) as x tends to infinity. What is the relation between the limit value you find and the horizontal asymptote?
 
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  • #3
horizontal asymptote: take limit of f(x) as x tends to infinity. What is the relation between the limit value you find and the horizontal asymptote?
I don't know but that answer looks a little too complicated...

y = (1-2x)/(3x+5)
3xy + 5y = 1 - 2x
x = (1 - 5y )/(3y + 2)

Now, find the value y can't have...

but beware, I have been wrong before (intentionally of course! )
 
  • #4
Originally posted by FZ+
I don't know but that answer looks a little too complicated...

y = (1-2x)/(3x+5)
3xy + 5y = 1 - 2x
x = (1 - 5y )/(3y + 2)

Now, find the value y can't have...

but beware, I have been wrong before (intentionally of course! )

That's interesting. I never thought of it like that. That suggests to me that the horizontal asymptote of a function should be equivalent to the vertical asymptote of its inverse (if the inverse exists). Let's see. That might be too general.

Say y(x)= (Ax^n+B)/(Cx^n+D)
Then y=A/C is its horizontal asymptote.
It's inverse:
yCx^n+yD=Ax^n+B
x^n(yC-A)=B-yD
x=[(B-yD)/(yC-A)]^(1/n)
Has as its vertical asymptote y=A/C
YEAH!
(For the second graph, x is a function of y - so y=A/C is a vertical line).

Of course, the converse should also be true.
I like that. It shows how arbitrary our placement of the axes and the definition of our variables are.
My algebra skills break down from there. I tried having nonzero coefficients for other powers of x [eg. x^(n-1)] but I'm not sure I can solve for x in that case.
 
  • #5
In this case the largest degrees of the variables are the same (to the first). So for horizontal asymtotes don't u just take the ratio:

-2x + 1
3x + 5

So the horizontal asymtote is (-2/3)

Right?
 
  • #6
right
To find horizontal asymptotes (when dealing with rational functions of polynomials), divide both top and bottom of the fraction by the highest power of x and take the limit as x->[oo]
Essentially, all terms which have an x that is less than the highest power will tend to zero, and that is a shortcut that most people use.

For example, if the highest power of x is in the denominator, all the terms in the numerator will tend to zero and that asymptote is y=0.

btw, an asymptote is a line (or some y-value) that the graph approaches, so it should be written as y=k, rather than just k, although I'm sure anyone would know what you mean if you said the horizontal asymptote is -2/3
 
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1. What is an asymptote?

An asymptote is a line that a curve approaches but never touches. It can be horizontal, vertical, or oblique.

2. How do you find the vertical asymptote of a rational function?

To find the vertical asymptote of a rational function, set the denominator equal to zero and solve for x. The resulting value(s) will be the x-coordinate(s) of the vertical asymptote.

3. What is the significance of the horizontal asymptote?

The horizontal asymptote represents the value that the function approaches as the input (x) gets larger and larger. It is also used to determine the end behavior of a function.

4. How do you graph a rational function with a slant asymptote?

To graph a rational function with a slant asymptote, first find the quotient using long division or synthetic division. The resulting quotient will be a polynomial function, which can be graphed as a slant asymptote. Then, plot the vertical and horizontal asymptotes and the points where the function intersects them. Finally, connect the points on the graph to create a curve that approaches the slant asymptote.

5. Can a function have multiple asymptotes?

Yes, a function can have multiple asymptotes. It can have both vertical and horizontal asymptotes, as well as oblique asymptotes. These asymptotes can intersect or be parallel to each other.

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