Finding horizontal and vertical asymptotes

In summary, the equation is x^2 + xy + y^2 = 25. The method used was to implicitly differentiate and set the numerator equal to 0 to find horizontal asymptotes and the denominator equal to 0 to find vertical asymptotes. However, upon using implicit differentiation, the resulting equation was y' = (-2x - y)/(x + 2y) and setting the numerator equal to 0 only gives y = -2x. This leaves the question of how to solve for the points where there are both horizontal and vertical asymptotes. After further discussion, it was determined that the only asymptotes for this expression are the oblique asymptotes y = x and y = -3x. The method used
  • #1
dnt
238
0
the equation is:

x^2 + xy + y^2 = 25

my method was to implicitely differentiate and then set the numerator = 0 to get points of horizontal asymptotes, and then set the denominator = 0 to get the points for the veritcal asymptotes.

when i use implicit differentiation i get:

y' = (-2x - y)/(x + 2y)

am i correct so far? now I am stuck because if i set the top equal to 0 i just get y = -2x. what can i do from here to solve for the points where there are both horizontal and vertical asymptotes? thanks.
 
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  • #2
Do you have any reason to think the graph of that expression has horizontal and vertical asymptotes? I get that the only asymptotes are the oblique asymptotes y= x and y= -3x.
 
  • #3
how can i know if it does have any to begin with?

and can you explain how you got those oblique asymptotes?
 

Related to Finding horizontal and vertical asymptotes

What is a horizontal asymptote and how do you find it?

A horizontal asymptote is a line that a graph approaches but never touches as the input values get increasingly larger or smaller. To find a horizontal asymptote, you can use the degree of the numerator and denominator of the rational 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 a vertical asymptote and how do you find it?

A vertical asymptote is a vertical line that a graph approaches but never crosses as the input values approach a certain value. To find a vertical asymptote, you can set the denominator of a rational function equal to zero and solve for the input value. The resulting input value will be the vertical asymptote.

Can a graph have multiple asymptotes?

Yes, a graph can have both horizontal and vertical asymptotes. It is also possible for a graph to have multiple horizontal or vertical asymptotes.

What does it mean if a graph has no asymptotes?

If a graph has no asymptotes, it means that the graph approaches a finite value as the input values get increasingly larger or smaller. In other words, the graph has a finite limit as the input values approach infinity or negative infinity.

How do you graph a rational function with horizontal and vertical asymptotes?

To graph a rational function with horizontal and vertical asymptotes, start by plotting the vertical asymptotes on the x-axis. Then, use the behavior of the function near the vertical asymptotes to plot its graph on either side. Next, find the horizontal asymptotes and plot them as dashed lines on the y-axis. Finally, plot additional points on the graph to ensure accuracy and sketch the curve between the asymptotes.

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