Final answer:
The value of a + b + 2c is 12.
Step-by-step explanation:
To find the value of a + b + 2c, we can use the information given in the problem.
1. We have the equation of the quadratic function as f(x) = ax² + bx + c. Since the parabola opens downward, the coefficient of a must be negative.
2. From the maximum point (-1,8), we can determine the values of a, b, and c. Plugging in the coordinates of the maximum point into the quadratic function, we get the equation -a + b + c = 8.
3. We also have the equation a + b + c = 8.
4. Adding these two equations, we get 2a + 2c = 16.
5. Rearranging this equation, we find a + c = 8.
6. Subtracting this equation from a + b + c = 8, we can solve for b. We find that b = 0.
7. Since a + c = 8, and a and c are equal, we can divide by 2 to find that a = c = 4.
8. Finally, we substitute the values of a = 4, b = 0, and c = 4 into the expression a + b + 2c. This gives us 4 + 0 + 2(4) = 12.