The required polynomial function that passes through the points (6,13), (9,-12), and (0,5) is y = 0.5x^2 - 7x + 5.
Step-by-step explanation:
To find a polynomial function whose graph passes through the points (6,13), (9,-12), and (0,5), we need at least a second degree polynomial since three points are given. The general form of a second-degree polynomial is y = ax2 + bx + c. Using the given points, we can set up a system of equations.
For point (0,5): c = 5
For point (6,13): 36a + 6b + c = 13
For point (9,-12): 81a + 9b + c = -12
Since we already know that c = 5 from the first point, we can substitute it into the other two equations and solve for a and b.
36a + 6b = 8
81a + 9b = -17
Solving this system, we find that:
a = 0.5
b = -7
Therefore, the polynomial function is y = 0.5x2 - 7x + 5.