Anytime it crosses or touches the x-axis, we have a zero.

Since it crosses the x-axis at 3, one of the factors of our equation must be x - 3 because (x - 3) = 0 at x = 3.

Since it touches the x-axis at -2, one of the factors of our equation must be x + 2 because (x + 2) = 0 at x = -2.

If we cross the x-axis, then the relevant factor has an odd power, whereas if we touch the x-axis, then the relevant factor has an even power.

Our equation could have (x - 3), (x - 3)^{3}, or (x - 3)^{5}, but not (x - 3)^{2}

Similarly, our equation could have (x + 2)^{2}, (x + 2)^{4}, or (x + 2)^{6}, but not (x + 2)

We have vertical asymptotes where the denominator is equal to 0. If we have a vertical asymptote at x = 1, then our denominator must contain as one of its factors x-1

We have horizontal asymptotes when the degree of the numerator is less than or equal to the degree of the denominator.

- If the degree is less, the asymptote is always y = 0
- If the degree is the same, the asymptote is the ratio of the leading coefficients (the coefficient of the numerator / the coefficient of the denominator)

Since the horizontal asymptote is y = 2, the degree of the numerator and denominator must be the same and the coefficients must have a ratio of 2.

So our equation could be:

- f(x) = (2(x - 3)(x + 2)
^{2}) / (x-1)^{3} - or f(x) = (2(x - 3)
^{3}(x + 2)^{2}) / (x-1)^{5} - or f(x) = (2(x - 3)(x + 2)
^{4}) / (x-1)^{5} - or f(x) = (2(x - 3)
^{11}(x + 2)^{22}) / (x-1)^{33} - or any other equation that follows the above conditions.