Asymptote

A asymptote It is a straight line that continuously approaches the curve of a function, but never touches it as the curve advances infinity in one direction, that is, the distance between the two tends to be zero, as they extend indefinitely.

Figure 2. Parametric equations of the curve
Figure 2. Parametric equations of the curve

Figure I shows the graphical representation of the function f (x) = 1 / x; the axes "x" and "y" are the asymptotes of the function. As x approaches infinity, the curve gets closer and closer to the x axis, but never touches it. Similarly, as the y-axis approaches infinity, the curve also gets closer and closer to the axis, but never touches it.

Procedure to find asymptotes:

Procedure 1:

  1. Substitute mx + c = y into the equation of the curve and match the coefficients of the two highest powers of x to zero.
  2. Determine the values of myca from the equations. If m1, c1, m2, c2, Y1= m1 x + c1 ; Y2 = m2 x + c2; … Yn = mn x + cn

Example: Find the asymptotes of the x-curves3 + 2x2 - (xy)2  - (2 and)3 + xy - y2 + 1 = 0

Substituting mx + c = y in the equation we obtain

x3 + 2x2 - [x (mx + c)]2 - [2 (mx + c)]3 + x (mx + c) - [(mx + c)]2 + 1 = 0

Equalizing the coefficients x to zero3 (higher grade), we have:

1 + 2m - m2 - 2m3 = 0 → (1 + 2m) (1 - m2) = 0

Thus:

m = 1; -1 ; -1/2

Equalizing the coefficients x to zero2, we have:

2c [1 - m - (3m)2] + m - m2 = 0

Thus:

m = 1 and c = 0; when m = -1 → c = -1; and when m = -1/2 → c = 1/2

The asymptotes will be:

y = x; y = - (x + 1); y = ½ - (x / 2)

Procedure 2 (Short method):

If we substitute x = 1 and y = m in the highest degree terms of the equation we obtain ϕn(m); ϕn-1(m) is obtained by substituting x = 1 and y = m in the (n-1) term. Equaling ϕn(m) = 0, we obtain the roots as m1, m2,… Mn. Values corresponding to c1, c2,… Cn is obtained by substituting using the formula:

cn= [ϕn-1(m)] / [ϕn(m)]

Therefore the asymptotes will be and1 = m1x + c1; Y2 = m2x + c2… Yn = mnx + cn

Example: Find the asymptotes of the x-curves3 + 2x2 - (xy)2 - (2 and)3 + xy - y2 + 1 = 0

Substituting x = 1 and y = m in the equation we obtain:

ϕ3(m) = 1 + 2m - m2 - 2m3      ; ϕ2(m) = m - m2