Answer
x-intercept: (-6,0) (1,0)
y-intercept: (0, 2)
Vertical Asymptote: x=-3, 2
Domain: (-∞, -3) U (-3, 2) U (2, ∞)
Horizontal Asymptote: y=2
Range: (-∞, ∞)
See graph below
Work Step by Step
$r(x)=\frac{2x^2 + 10x - 12}{(x^2 + x - 6)}$
$2x^2 + 10x - 12 = 0$
$2 (x^2 + 5x - 6) = 0$
$2 (x+6)(x-1) = 0$
x = -6, 1
x-intercept: (-6,0) (1,0)
y-intercept is the ratio of the constants, which is -12 / -6 = 2
Thus, the y-intercept is at (0, 2)
Vertical asymptotes are when the denominator is equal to 0
$x^2 + x - 6 = 0$
$(x+3)(x-2) = 0$
x = -3, 2
So, domain is from (-∞, -3) U (-3, 2) U (2, ∞)
Horizontal asymptote is the ratio of the constants of the leading term (with equal degree)
2/1 = 2
Thus, the horizontal asymptote is at y=2
So the range is from (-∞, ∞)