Graph all vertical and horizontal asymptotes of the rational function.
F(x)=x^2-4x+3/-x^2+3x+4
To find the vertical asymptotes of a rational function, we need to determine the values of x where the denominator of the function is equal to zero.
For the given function, F(x) = (x^2 - 4x + 3) / (-x^2 + 3x + 4), we set the denominator equal to zero and solve for x:
-x^2 + 3x + 4 = 0
This is a quadratic equation that can be factored as:
-(x - 4)(x + 1) = 0
Therefore, x = 4 and x = -1 are the vertical asymptotes of the function.
To find the horizontal asymptote, we need to determine the behavior of the function as x approaches positive or negative infinity. We can do this by comparing the degrees of the numerator and denominator of the function.
F(x) = (x^2 - 4x + 3) / (-x^2 + 3x + 4)
The degrees of the numerator and denominator are both 2, so we look at the leading coefficients of both:
As x approaches infinity:
The leading coefficient of the numerator is 1 and the leading coefficient of the denominator is -1. Therefore, there is no horizontal asymptote.
In summary, the vertical asymptotes of the rational function F(x) = (x^2 - 4x + 3) / (-x^2 + 3x + 4) are x = 4 and x = -1, and there is no horizontal asymptote.