Answer
x-intercept: None
y-intercept: (0, -2)
Vertical Asymptote: x=-1, 3
Domain: (-∞, -1) U (-1, 3) U (3, ∞)
Horizontal Asymptote: y=3
Range: (-∞, 3) U (3, ∞)
See graph below
Work Step by Step
$r(x)=\frac{3x^2 + 6}{(x^2 -2x - 3)}$
$3x^2 + 6 = 0$
$3(x^2 + 2) = 0$
No x-intercept
y-intercept is the ratio of the constants, which is 6/-3 = -2
Thus, the y-intercept is at (0, -2)
Vertical asymptotes are when the denominator is equal to 0
$x^2 -2x - 3 = 0$
$(x-3)(x+1) = 0$
x = 3, -1
So, domain is from (-∞, -1) U (-1, 3) U (3, ∞)
Horizontal asymptote is the ratio of the constants of the leading term (with equal degree)
3/1 = 3
Thus, the horizontal asymptote is at y=3
So the range is from (-∞, 3) U (3, ∞)