Write an equation for a rational function with: Vertical asymptotes at x = - 3 and x = - 4 x-intercepts at x = 3 and x = 4 Horizontal asymptote at y = 10

(10x ^ 2 - 70 * x + 120) / (x ^ 2 + 7 * x + 12)

Step-by-step explanation:

First a rational function is a function that is in the form a / b.

Knowing this, we proceed to calculate the asymptotes in x.

Vertical asymptotes

So that there is an asymptote in x, in a rational function the easiest thing is that the denominator is 0. Therefore, for the asymptotes x = - 3 and x = - 4, we make it 0 in both cases. That is to say:

x + 3 = 0

x + 4 = 0

both, in the denominator. For the moment it would be something like this:

1 / (x + 3) * (x + 4)

X Intersections

Now, for intersections, what we should do equation 0, and that the solution of x us of the value we want, in this case x = 3 and x = 4. Because here the denominator does not influence because when equal to 0 to a rational function, the denominator becomes 0. Similarly, it is equal to 0 as in the previous case, only this time they would go in the numerator. It would be:

x - 3 = 0

x - 4 = 0

Therefore, the function would go like this: (x - 3) * (x - 4) / (x + 3) * (x + 4),

Operating the following we have:

(x ^ 2 - 4 * x - 3 * x + 12) / (x ^ 2 + 4 * x + 3 * x + 12)

(x ^ 2 - 7 * x + 12) / (x ^ 2 + 7 * x + 12)

having asymptotes at x = - 3 and x = - 4, and intersection at (3.0) and (4.0), that is x = 3 and x = 4.

Horizontal asymptote.

Finally, in the case of the horizontal asymptote, which must be y = 10, what must be done is to divide all the terms by the highest degree term, in this case it is x ^ 2. It would be as follows:

(x ^ 2 / x ^ 2 - 7 * x / x ^ 2 + 12 / x ^ 2) / (x ^ 2 / x ^ 2 + 7 * x / x ^ 2 + 12 / x ^ 2)

Now, x ^ 2 / x ^ 2 = 1, and all the others have to be 0, therefore it would remain (1 + 0 + 0) / (1 + 0 + 0) = 1

So for the asymptote to be 10, the entire denominator must be multiplied by 10, thus being:

10 * (x ^ 2 - 7 * x + 12) / (x ^ 2 + 7 * x + 12)

(10x ^ 2 - 70 * x + 120) / (x ^ 2 + 7 * x + 12)

And this equation would already comply with everything required in the problem.