Questions: The figure below shows the graph of a rational function f. It has vertical asymptotes x=5 and x=-6, and horizontal asymptote y=-3. The graph has x-intercepts 1 and -3, and it passes through the point (3,2). The equation for f(x) has one of the five forms shown below. Choose the appropriate form for f(x), and then write the equation. You can assume that f(x) is in simplest form. f(x)=a/(x-b) f(x)=a(x-b)/(x-c) f(x)=a/((x-b)(x-c)) f(x)=a(x-b)/((x-c)(x-d)) f(x)=a(x-b)(x-c)/((x-d)(x-e))

Solution Steps

The graph has vertical asymptotes at x = 5 and x = -6, x-intercepts at x = 1 and x = -3, and a horizontal asymptote at y = -3. The graph also passes through the point (3, 2).

Since there are two vertical asymptotes and two x-intercepts, the denominator must have two factors and the numerator must also have two factors. The horizontal asymptote is not y=0, so the degree of the numerator must be equal to the degree of the denominator. Therefore, the general form is:

f(x) = a(x - b)(x - c) / (x - d)(x - e)

where a, b, c, d, and e are constants.

The vertical asymptotes are at x = 5 and x = -6, so the denominator is (x - 5)(x + 6). The x-intercepts are at x = 1 and x = -3, so the numerator is (x - 1)(x + 3). The general form of the equation is:

f(x) = a(x - 1)(x + 3) / (x - 5)(x + 6)

Since the graph passes through the point (3, 2), we can substitute x = 3 and f(x) = 2:

2 = a(3 - 1)(3 + 3) / (3 - 5)(3 + 6)
2 = a(2)(6) / (-2)(9)
2 = 12a / -18
-36 = 12a
a = -3

Final Answer:

f(x) = -3(x - 1)(x + 3) / (x - 5)(x + 6)