Answer:
x = -3 and x = 5, x-intercepts at (-5, 0) and (2, 0), and a horizontal asymptote at y = -2.
Step-by-step explanation:
To write an equation for a rational function with the given characteristics, we can start by considering the asymptotes and x-intercepts.
The vertical asymptotes at x = -3 and x = 5 indicate that the denominator of the rational function must have factors of (x + 3) and (x - 5), respectively.
The x-intercepts at (-5, 0) and (2, 0) indicate that the numerator of the rational function must have factors of (x + 5) and (x - 2), respectively.
To ensure that the rational function has a horizontal asymptote at y = -2, the degree of the numerator must be less than or equal to the degree of the denominator.
Now, let's put all this information together to write the equation for the rational function:
f(x) = (x + 5)(x - 2) / ((x + 3)(x - 5))
This equation represents a rational function with vertical asymptotes at x = -3 and x = 5, x-intercepts at (-5, 0) and (2, 0), and a horizontal asymptote at y = -2.
