Final answer:
The equation of the quadratic with roots 1 and 5, and a value of -9 when x is 7, is f(x) = -3/4(x - 1)(x - 5).
Step-by-step explanation:
To write the equation of the quadratic with the given roots and a specific function value, you can use the fact that if a quadratic function f(x) has roots 1 and 5, the function can be expressed in the form f(x) = a(x - 1)(x - 5), where a is a coefficient we need to determine. We are also given that f(7) = -9. By substituting x = 7 into the function, you get -9 = a(7 - 1)(7 - 5) = a(6)(2). Simplifying, you find that -9 = 12a, so a = -9/12 = -3/4. Thus, the equation of the quadratic is f(x) = -3/4(x - 1)(x - 5).