Write a rational function that has the following characteristics: (There could be more than 1 possible answer)

Question

1. Crosses the x-axis at 3
2. touches the x-axis at -2
3. has a vertical asymptote at x=1
4. has a horizontal asymptote at y=2

Answer 1

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.

1. If the degree is less, the asymptote is always y = 0
2. 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:

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