Answer:
To find the x-intercept, we set y = 0 and solve for x:
r(x) = 2x^2 + 10x - 12 / x^2 + x - 6
0 = (2x^2 + 10x - 12) / (x^2 + x - 6)
0 = 2(x-1)(x+6) / (x-2)(x+3)
This gives us x-intercepts of x = 1 and x = -6.
To find the y-intercept, we set x = 0 and solve for y:
r(0) = 2(0)^2 + 10(0) - 12 / (0)^2 + (0) - 6
r(0) = -12/-6 = 2
So the y-intercept is y = 2.
To find the vertical asymptotes, we set the denominator of the function equal to zero and solve for x:
x^2 + x - 6 = 0
(x+3)(x-2) = 0
This gives us vertical asymptotes at x = -3 and x = 2.
To find the horizontal asymptote, we look at the degrees of the numerator and denominator. Since the degree of the numerator is 2 and the degree of the denominator is also 2, we divide the leading coefficient of the numerator by the leading coefficient of the denominator:
2 / 1 = 2
So the horizontal asymptote is y = 2.
Therefore, the x-intercepts are x = 1 and x = -6, the y-intercept is y = 2, the vertical asymptotes are x = -3 and x = 2, and the horizontal asymptote is y = 2.
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