How to write an equation for a rational function with: vertical asymptotes at x = 3 and x = -5?

Many functions will work but here is the simplest one:

graph{1/(x^2+2x-15) [-10, 10, -5, 5]}

To write an equation for a rational function with vertical asymptotes at ( x = 3 ) and ( x = -5 ), you can use the factors of the denominator to determine the behavior of the function near these asymptotes. The general form of a rational function is ( f(x) = \frac{p(x)}{q(x)} ), where ( p(x) ) and ( q(x) ) are polynomials.

To have vertical asymptotes at ( x = 3 ) and ( x = -5 ), the denominator ( q(x) ) must have factors of ( (x - 3) ) and ( (x + 5) ), respectively. Thus, the denominator could be ( q(x) = (x - 3)(x + 5) ) or any multiple of this expression.

For example, one possible equation for the rational function could be ( f(x) = \frac{1}{(x - 3)(x + 5)} ). This function will have vertical asymptotes at ( x = 3 ) and ( x = -5 ).