Парабола y=ax^2+bx+c имеет вершину в точке M (2; 1) и проходит через точку K (-1; 5). Найдите значение коэффициентов a, b и c.

\[вершина\ \ M\ (2;1);\ \ \]

\[проходит\ через\ \ K\ ( - 1;5).\]

\[1)\ x_{0} = \frac{- b}{2a}\text{\ \ \ \ \ \ }\]

\[\frac{- b}{2a} = 2\]

\[b = - 4a.\]

\[y_{0} = ax^{2} - 4ax + c\]

\[4a - 8a + c = 1\]

\[- 4a + c = 1.\]

\[2)\ a - b + c = 5\]

\[3)\ \left\{ \begin{matrix} b = - 4a\ \ \ \ \ \ \ \ \\ - 4a + c = 1\ \ \\ a - b + c = 5 \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} 4a + b = 0\ \ \ \ \ \ \ \ \\ - 4a + c = 1\ \ \ \ \ \\ a + 4a + c = 5 \\ \end{matrix}\ ( + )\text{\ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} b = - 4a\ \ \ \ \ \ \ \ \ \\ - 4a + c = - 1 \\ 5a + c = 5\ \ \ \ \ \ \\ \end{matrix} \right.\ ( - )\]

\[\left\{ \begin{matrix} b = - 4a\ \ \ \ \ \ \ \\ - 9a = - 4\ \ \ \ \\ - 4a + c = 1 \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} a = \frac{4}{9}\text{\ \ \ \ \ \ \ \ \ \ \ } \\ b = - \frac{16}{9}\text{\ \ \ \ \ } \\ c = 1 + \frac{16}{9} \\ \end{matrix}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} a = \frac{4}{9}\text{\ \ \ \ \ } \\ b = - 1\frac{7}{9} \\ c = 2\frac{7}{9} \\ \end{matrix} \right.\ \]

\[Ответ:a = \frac{4}{9};\ \ b = - 1\frac{7}{9};\ \ \]

\[c = 2\frac{7}{9}.\]