Maximum Height of Thrown Ball

Consider the function $$f(x) = ax^2 + bx + c$$ which models the height of a ball thrown upward from an initial height of $$c$$ with initial velocity $$v$$ and subject to gravity. The coefficients are defined as follows:

• $$a = -rac{g}{2}$$
• $$b = v$$
• $$c = h_0$$

where $$g$$ represents the acceleration due to gravity, $$v$$ is the initial velocity, and $$h_0$$ is the initial height. Assuming that $$g = 9.8$$ m/s², the ball is thrown from an initial height of $$h_0 = 10$$ meters with an initial velocity of $$v = 20$$ m/s. Calculate the maximum height that the ball will reach.