Since the function describing the height is a quadratic function with negative leading coefficient this means that this is a parabola that opens down. This also means that the maximum height will be given as the y component of the vertex of the parabola, then if we want to find the maximum height, we need to write the function in vertex form so let's do that:
[tex]\begin{gathered} h(t)=-4.9t^2+19t+1.5 \\ =-4.9(t^2+\frac{19}{4.9}t)+1.5 \\ =-4.9(t^2+\frac{19}{4.9}t+(\frac{19}{9.8})^2)+1.5+4.9(\frac{19}{9.8})^2 \\ =-4.9(t+\frac{19}{9.8})^2+19.9 \end{gathered}[/tex]Hence the function can be written as:
[tex]h(t)=-4.9(t+1.9)^2+19.9[/tex]and its vertex is at (1.9,19.9) which means that the maximum height of the ball is 19.9 m
