First, we factor out the numerator and denominator completely.
f(x) = [t(t2 - 2t - 8)] / [(t - 2)(t + 2)] =
f(x) = [t(t - 4)(t + 2)] / [(t - 2)(t + 2)]
To find the y-intercept, evaluate f(0).
To find the hole, find a factor that is in the numerator and denominator. From this function, you will see that (t + 2) is a factor that is both in the numerator and denominator. In other words, these two factors cancel each other out. Therefore, a hole occurs at t=-2.
To find the x-intercept, set the numerator equal to zero and solve for t. Recall that we cancelled out a term in the earlier step. So now the function becomes
f(x) = t(t - 4) / (t - 2)
Now you can set the numerator equal to zero and solve for t to find the x-intercepts.
To find the vertical asymptotes, set the denominator equal to zero and solve for t.
There is no horizontal asymptote since the degree of the numerator is greater than the degree of the denominator.