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 asymptot

Question

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

in progress 0
Samantha 4 days 2021-10-12T12:42:44+00:00 1 Answer 0

Answers ( )

    0
    2021-10-12T12:44:22+00:00

    Answer:

    (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 Р3) * (x Р4) / (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.

Leave an answer

45:7+7-4:2-5:5*4+35:2 =? ( )