Rational functions

Question

Write a rational function with a vertical asymptote

x=-2 and 2 and a horizontal asymptote y=4

Answer 1

Another example is y = 4x^2 / (x^2 - 4)

If the degree of the numerator (largest power of x) is less than the degree of the denominator, then the horizontal asymptote is at y = 0. Adding 4 would shift it to y=4,

y = (1 / (x^2 - 4^2) ) + 4

If the degrees of the numerator and denominator are equal, then there is a horizontal asymptote at a/b where a is the coefficient (number in front of) the highest power of x on top, and b is the coefficient of the highest power of x on the bottom.

y = 4x^2 / (x^2 - 4)

Answer 2

To get a vertical asymptote at x = -2, use (x+2) in the denominator. This becomes zero when x = -2 and causes f(x) to approach infinity. 1/(x+2) is a perfect example.

To get a vertical asymptote at x = 2, use (x-2) in the denominator. This becomes zero when x = 2 and causes f(x) to approach infinity. 1/(x-2) is a perfect example.

If

f(x) = { 1 / [ ( x - 2 ) * ( x + 2 ) ] } + 4

then as x approaches infinity,

1 / [ ( x - 2 ) * ( x + 2 ) ] = 1 / ( x² - 4 )

approaches zero and thus f(x) approaches 4.

This creates a horizontal asymptote at y = 4.