Answer
x-intercept: None
y-intercept: (0, -9/8)
Vertical Asymptote: x=4
Domain: (-∞, 4) U (4, ∞)
Horizontal Asymptote: y=-1
Range: (-∞, -1)
See graph below
Work Step by Step
$r(x)=\frac{-x^2 + 8x - 18}{x^2 - 8x + 16}$
$-x^2 + 8x - 18= 0$
$x^2 - 8x + 18 = 0$
$x^2 - 8x + 16 = -18 + 16$
No x-intercept
y-intercept is the ratio of the constants, which is -18/16 = -9/8
Thus, the y-intercept is at (0, -9/8)
Vertical asymptotes are when the denominator is equal to 0
$x^2 - 8x + 16 = 0$
$(x-4)^2 = 0$
x = 4
So, domain is from (-∞, 4) U (4, ∞)
Horizontal asymptote is the ratio of the constants of the leading term (with equal degree)
-1/1 = -1
Thus, the horizontal asymptote is at y=-1
So the range is from (-∞, -1)
