Final answer:
To find a polynomial function that passes through the given points, assume the function has the form f(x) = ax^2 + bx + c. Substitute the x and y values from each point into the function, solve a system of equations, and find the values of a, b, and c. The polynomial function that passes through (5,12), (9,-11), and (0,5) is f(x) = -2x^2 + 23x + 5.
Step-by-step explanation:
To find a polynomial function that passes through the given points, we need to use the fact that polynomial functions can be written as a sum of monomials. Let's assume the function has the form: f(x) = ax^2 + bx + c. We can then substitute the x and y values from each point into the function and solve a system of equations to find the values of a, b, and c.
Using the point (5, 12), we get: 12 = a(5^2) + b(5) + c. Using the point (9, -11), we get: -11 = a(9^2) + b(9) + c. And using the point (0, 5), we get: 5 = a(0^2) + b(0) + c. We now have a system of three equations with three unknowns.
Solving this system of equations gives us the values: a = -2, b = 23, c = 5. Therefore, the polynomial function that passes through the given points is: f(x) = -2x^2 + 23x + 5.