Answer
x-intercept: (0, 0)
y-intercept: (0, 0)
Vertical Asymptote: x=-1, 3
Domain: (-∞, -1) U (-1, 3) U (3, ∞)
Horizontal Asymptote: y=4
Range: (-∞, 0] U [3, ∞)
See graph below
Work Step by Step
$r(x)=\frac{4x^2}{(x^2 -2x -3)}$
$4x^2 = 0$
x = 0
x-intercept: (0,0)
y-intercept is the ratio of the constants, which is 0/-3 = 0
Thus, the y-intercept is at (0, 0)
Vertical asymptotes are when the denominator is equal to 0
$x^2 -2x - 3= 0$
$(x+1)(x-3) = 0$
x = -1, 3
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)
4/1 = 4
Thus, the horizontal asymptote is at y=4
So the range is from (-∞, 0] U [3, ∞) (according to the graph)
