Final answer:
The equation Y = (x-2)/(x² -6x+8) has a hole at x = 2 after simplification. There is no root or vertex as it does not produce a parabola.
Step-by-step explanation:
The equation Y = (x-2)/(x² -6x+8) may either have a hole or an asymptote. To determine which one, we should factor the denominator and look for common factors in the numerator and denominator. The denominator factors to (x-2)(x-4). If there is a common factor in the numerator, this will result in a hole at that x-value. Since the numerator is also (x-2), we can simplify the equation, which means there is a hole at x = 2 because the factor (x-2) is canceled out. There's no x-intercept (root) because after canceling (x-2), the remaining function is 1/(x-4) which does not have a zero. The equation does not represent a parabola, so it does not have a vertex.