Quandaries
and Queries
What are the equations of all horizontal and vertical asymptotes for the curve y=x/(x(x2-4))(the answer is y=0, x=-2, x=2, but I want to know how to get that algebraically.And why isn't x=0 another asymptote?)
Hi Abraham.
To find the equations of the horizontal (or other type, like slant or parabolic) asymptote for a rational function, we examine the degrees of the polynomials in the numerator and denominator:
- If the degree of the numerator is less than the degree
of the denominator, then there is a horizontal asymptote at y=0.
- If the degrees are equal, then there is a horizontal
asymptote at y=a/b where a is the leading coefficient of the numerator
and b is the leading coefficient of the denominator.
- If the degree of the numerator is greater than the degree of the denominator, then the equation of the asymptote will not be horizontal, it will be a slanted line if the difference in degrees is 1, parabolic if the difference in degrees is 2, and so on. We find these equations by performing polynomial division (numerator divided by denominator) and the quotient (ignore any remainder) is the equation of the asymptote.
As for the vertical asymptotes, you set the denominator equal to zero and solve for the zeroes of the function. I believe in this case that x=0 is not an asymptote because the rational function can be reduced to y=1/(x2-4) and so there is no x factor left in the denominator. The factors from x2-4, namely x-2 and x+2 are used to solve for the two vertical asymptotes by setting the factors to zero and solving: x=2 and x=-2.
Hope this helps,
Leeanne