Write an equation for a rational function with:
Vertical asymptotes at x=-1 and x=4
x intercepts at x=-6 and x=6
Horizontal asymptote at y=2
y=
To find an equation for a rational function with the given characteristics, we can start by identifying the factors of the function.
Since there are vertical asymptotes at x=-1 and x=4, the factors corresponding to these asymptotes are (x+1) and (x-4), respectively.
As for the x-intercepts at x=-6 and x=6, the factors associated with these intercepts are (x+6) and (x-6), respectively.
Next, we need to determine the horizontal asymptote at y=2. This indicates that the degrees of the numerator and denominator of the rational function must be the same. In this case, since the horizontal asymptote is a straight line, the degrees of the numerator and denominator should be equal to zero.
Now we can construct the equation:
Start with y = a(x+1)(x-4)(x+6)(x-6) / b(x+1)(x-4),
where "a" and "b" are constants yet to be determined. We include (x+1)(x-4) in both the numerator and denominator to account for the vertical asymptotes.
Since the numerator should have a degree of zero, we have (x+1)(x-4)(x+6)(x-6) = a.
Similarly, the denominator should have a degree of zero, resulting in (x+1)(x-4) = b.
Now, we'll simplify these equations:
For the numerator:
(x+1)(x-4)(x+6)(x-6) = a
[(x^2-3x-4)(x^2-36)] = a
(x^4 - 39x^2 + 144) = a
For the denominator:
(x+1)(x-4) = b
(x^2-3x-4) = b
Since the denominators are the same, we can simplify the equation by canceling out these factors:
y = a(x^4 - 39x^2 + 144) / b(x^2-3x-4).
Finally, since we are given that the horizontal asymptote is at y=2, we can set the leading coefficients of the numerator and denominator equal to determine the values of "a" and "b":
2 = a/b.
Hence, the final rational function equation is:
y = 2(x^4 - 39x^2 + 144) / (x^2-3x-4).