Suppose we have a quadratic function $y = a x ^ { 2 } + b x + c$ in $x$ which satisfies the following conditions 【＊】： 【＊】 When $x = - 1$ ，then $y = - 8$ and when $x = 3$ ，then $y = 16$ ．Further，in the interval $- 1 \leqq x \leqq 3$ ，the value of $y$ increases with the increase of the value of $x$ ． We are to find the conditions which $a , b$ and $c$ must satisfy． From【＊】，it follows that $b$ and $c$ can be expressed in terms of $a$ as $$\begin{aligned}
& b = \mathbf { A B } a + \mathbf { A } \\
& c = \mathbf { D E } a - \mathbf { F } .
\end{aligned}$$ Hence，the axis of symmetry of the graph of this quadratic function has the equation $$x = \mathbf { G } - \frac { \mathbf { H } } { a } .$$ Thus $a , b$ and $c$ must satisfy the relationships（1）and（2），and furthermore $$0 < a \leqq \frac { \mathbf { I } } { \mathbf { J } } \quad \text { or } \frac { \mathbf { K L } } { \mathbf { M } } \leqq a < 0 .$$
Suppose we have a quadratic function $y = a x ^ { 2 } + b x + c$ in $x$ which satisfies the following conditions 【＊】：
【＊】 When $x = - 1$ ，then $y = - 8$ and when $x = 3$ ，then $y = 16$ ．Further，in the interval $- 1 \leqq x \leqq 3$ ，the value of $y$ increases with the increase of the value of $x$ ．
We are to find the conditions which $a , b$ and $c$ must satisfy．
From【＊】，it follows that $b$ and $c$ can be expressed in terms of $a$ as
$$\begin{aligned}
& b = \mathbf { A B } a + \mathbf { A } \\
& c = \mathbf { D E } a - \mathbf { F } .
\end{aligned}$$
Hence，the axis of symmetry of the graph of this quadratic function has the equation
$$x = \mathbf { G } - \frac { \mathbf { H } } { a } .$$
Thus $a , b$ and $c$ must satisfy the relationships（1）and（2），and furthermore
$$0 < a \leqq \frac { \mathbf { I } } { \mathbf { J } } \quad \text { or } \frac { \mathbf { K L } } { \mathbf { M } } \leqq a < 0 .$$