Determine the holes, vertical asymptotes and horizontal asymptotes of the rational function y=(3x^(2)+8x-10)/(x^(2)+7x+12)

Hole:
Vertical Asymptote:
Horizontal Asymptote:

y = (3x^2+8x-10)/(x^2+7x+12)

= (3x^2+8x-10) / (x+3)(x+4)

no holes, since y is never 0/0

vertical asymptotes where the denominator is zero: x = -3,-4

as x gets huge, y -> 3x^2/x^2 = 3
so that is the horizontal asumptote

http://www.wolframalpha.com/input/?i=(3x%5E2%2B8x-10)+%2F+(x%5E2%2B7x%2B12)+for+-10+%3C%3D+x+%3C%3D+10

To determine the holes, vertical asymptotes, and horizontal asymptotes of the rational function y = (3x^2 + 8x - 10) / (x^2 + 7x + 12), we need to analyze the behavior of the function as x approaches certain values.

Hole:
To find any possible holes in the function, we need to determine if there are any common factors between the numerator and the denominator that can be cancelled out. In this case, there are no common factors between 3x^2 + 8x - 10 and x^2 + 7x + 12. Therefore, there are no holes in the function.

Vertical Asymptotes:
Vertical asymptotes occur when the denominator of a rational function equals zero. To find the vertical asymptotes, we need to solve the equation x^2 + 7x + 12 = 0 for x. This can be factored as (x + 3)(x + 4) = 0. Thus, the solutions are x = -3 and x = -4. These values represent the vertical asymptotes of the function.

Horizontal Asymptotes:
To find the horizontal asymptotes, we need to compare the degrees of the numerator and the denominator of the rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = leading coefficient of the numerator divided by the leading coefficient of the denominator. Finally, if the degree of the numerator is greater than the degree of the denominator, the rational function does not have a horizontal asymptote.

In this case, the degree of the numerator is 2 and the degree of the denominator is also 2. Therefore, the horizontal asymptote is y = leading coefficient of the numerator (3) divided by the leading coefficient of the denominator (1). So, the horizontal asymptote is y = 3.

To summarize:
- There are no holes in the rational function.
- The vertical asymptotes are x = -3 and x = -4.
- The horizontal asymptote is y = 3.