Question

I need to write an equation for a rational function here is a picture horizontal asymptotes and vertical asymptotes and X intercepts are in the photo

Answer 1

Given

Vertical asymptotes at x=1 and x=-6.

x-intercepts at x=4 and x=-3.

Horizontal asymptote at y=3.

To write the rational function.

Now,

Since the vertical asymptotes are at x=1 and x=-6.

Then, the denominator of the rational function is (x-1)(x+6).

Also, the intercepts are at x=4 and x=-3.

Then, the numerator of the rational function is (x-4)(x+3).

That implies,

$\begin{gathered} f(x)=\frac{a(x-4)(x+3)}{(x-1)(x+6)} \\ \end{gathered}$

Since the horizontal asymptote y=3.

Then, the function of x approaches infinity.

That is, when x=1, then f(x) will be infinity.

Therefore,

$f(x)=\frac{(x-4)(x+3)}{(x-1)(x+6)}$