Final answer:
To find the quadratic function, write it in factored form with x-intercepts at -1 and 5, then find the leading coefficient to ensure the vertex is the maximum at y=9, resulting in the function f(x) = -3(x + 1)(x - 5).
Step-by-step explanation:
To find a quadratic function with x-intercepts of -1 and 5 and a range of  (∞infinity,9–), we first use the factored form of a quadratic equation, which is f(x) = a(x - r1)(x - r2), where r1 and r2 are the roots or x-intercepts of the function. In this case, r1 = -1 and r2 = 5. Thus, the function has a form of f(x) = a(x + 1)(x - 5). Since the range tops out at 9, this value is the maximum value of the function, implying that the parabola opens downwards. Therefore, coefficient a must be negative.
The vertex of the function can be found halfway between the x-intercepts, at x = (5 - 1)/2 = 2. To ensure that the vertex is the highest point on the parabola, we calculate the y-value at the vertex using the x-intercepts. Plugging in x = 2 into f(x) = a(x + 1)(x - 5), we set f(2) = 9 to find a. This leads to the equation 9 = a(2 + 1)(2 - 5), which simplifies to 9 = -3a. Solving for a gives us a = -3. Therefore, the quadratic function is f(x) = -3(x + 1)(x - 5).