how do i find all vertical and horizontal asymptotes for the function y= 3x^2-8x+25/x^2+7x-18
vertical asymptotes result when the denominator is zero, so
x^2 + 7x - 18 = (x+9)(x-2)
vertical asymptotes: x = -9 and x = 2
for horizontal asymptotes let x -->∞
then y ---> 3/1 = 3
Horizontal asymtote: y = 3
To find the vertical and horizontal asymptotes of the function y = (3x^2 - 8x + 25) / (x^2 + 7x - 18), we need to follow a few steps.
1. Determine vertical asymptotes:
Vertical asymptotes occur when the denominator of the rational function equals zero. In this case, solve the equation x^2 + 7x - 18 = 0 for x to find the values that make the denominator zero. The solutions to this equation will be the x-values where the vertical asymptotes exist.
In the given equation, x^2 + 7x - 18 = 0 can be factored as (x - 2)(x + 9) = 0. Set each factor equal to zero:
x - 2 = 0 --> x = 2
x + 9 = 0 --> x = -9
Therefore, the vertical asymptotes are x = 2 and x = -9.
2. Determine horizontal asymptotes:
Horizontal asymptotes can be found by analyzing the degrees of the numerator and denominator of the rational function. The degrees provide information about the behavior of the function as x approaches positive or negative infinity.
Compare the degrees of the numerator and denominator:
Degree of numerator = 2
Degree of denominator = 2
In this case, since both the numerator and denominator have the same degree, we look at the ratio of the leading coefficients. The leading coefficients are the coefficients of the highest degree terms.
The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. The ratio of these coefficients is 3/1 = 3.
Therefore, the horizontal asymptote is y = 3.
So, the vertical asymptotes are x = 2 and x = -9, and the horizontal asymptote is y = 3.