Final answer:
To determine the value of c in the quadratic equation f(x), we need the values of a and b, which aren't provided. With only one value of f(x), we cannot solve for c without additional information.
Step-by-step explanation:
To find the value of the constant c in the quadratic equation f(x) = ax² + bx + c, we can use the given points, where the function f(x) equals to specific values at certain x values.
We have the points:
- f(2) = 3
- f(5) = -6
- f(7) = 10
By substituting these into the function, we'll have three equations:
- a(2)² + b(2) + c = 3
- a(5)² + b(5) + c = -6
- a(7)² + b(7) + c = 10
To solve for c, we only require one equation. The easiest way to do this is by using the first point. Plugging in x = 2, we get:
4a + 2b + c = 3 (1)
We don’t know the values of a and b, but we know the equation must hold true for any values of a and b due to the points given. Therefore, we can't solve for c without two more equations to find a and b. By creating similar equations with the other points provided, we can solve for a, b, and c simultaneously.