\(\begin{array}{l}A(0,1) \Rightarrow 1 = a{(0)^2} + b(0) + c \Rightarrow c = 1\\\\B( - 1,0) \Rightarrow 0 = a{( - 1)^2} + b( - 1) + 1 \Rightarrow a - b = - 1\\\\C(4,0) \Rightarrow 0 = a{(4)^2} + b(4) + 1 \Rightarrow 16a + 4b = - 1\\\\4 \times \left\{ \begin{array}{l}a - b = - 1\\\\16a + 4b = - 1\end{array} \right. \Rightarrow \left\{ \begin{array}{l}4a - 4b = - 4\\\\16a + 4b = - 1\end{array} \right. \Rightarrow 20a = - 5\\\\ \Rightarrow a = - \frac{1}{4},b = \frac{3}{4}\\\\y = - \frac{1}{4}{x^2} + \frac{3}{4}x + 1\end{array}\)