Consider two quadratic functions
$$\begin{aligned} & y = 2 x ^ { 2 } + 3 a x + 4 b \tag{1}\\ & y = b x ^ { 2 } + c x + d \tag{2} \end{aligned}$$
whose graphs are mutually symmetric with respect to the origin.
(1) From the symmetry with respect to the origin we see that
$$b = \mathbf { AB } , \quad c = \mathbf { C } a , \quad d = \mathbf { D } .$$
Hence (2) can be reduced to
$$y = \mathbf{AB} x ^ { 2 } + \mathbf{C} a x + \mathbf{D} . \tag{3}$$
(2) Let $0 < a < 1$, and consider the graph of (3).
When the range of values of $x$ is $0 \leqq x \leqq \frac { 3 } { 2 }$, the range of values of $y$ in (3) is
$$\frac { \mathbf { E } } { \mathbf { F } } a ^ { 2 } - \frac { \mathbf { G } } { \mathbf { H } } \leq y \leqq \frac { \mathbf { I } } { \mathbf{J}} a ^ { 2 } + \mathbf { K }$$
(3) For any value of $a$, the vertex of the graph of (3) is on the graph of the quadratic function
$$y = \mathbf { L } x ^ { 2 } + \mathbf { M } .$$

Let $a \neq 0$. Let $G$ be a curve which is symmetric with respect to the origin $(0,0)$ to the graph of the quadratic function in $x$
$$y = ax^2 - 4x - 4a. \tag{1}$$
(1) The coordinates of the vertex of the graph of (1) are
$$\left( \frac{\mathbf{A}}{a}, -\frac{\mathbf{B}}{a} - 4a \right).$$
(2) Among the following choices, the quadratic function whose graph is $G$ is $\square$ C. (0) $y = ax^2 + 4x + 4a$
(1) $y = ax^2 + 4x - 4a$
(2) $y = ax^2 - 4x + 4a$
(3) $y = -ax^2 + 4x + 4a$
(4) $y = -ax^2 - 4x + 4a$
(5) $y = -ax^2 - 4x - 4a$
(3) The curve $G$ intersects the graph of the quadratic function (1) at the two points
$$(\mathrm{DE},\ \mathrm{F}) \text{ and } (\mathrm{G},\ \mathrm{HI}).$$
(4) Let $a = 2$. Then over the interval $\mathrm{DE} \leqq x \leqq \square$, the maximum and the minimum values of the quadratic function whose graph is $G$ are $\square$ JK and $\square$ LM, respectively.

Let $a \neq 0$. Let $G$ be a curve which is symmetric with respect to the origin $(0,0)$ to the graph of the quadratic function in $x$
$$y = ax^2 - 4x - 4a. \tag{1}$$
(1) The coordinates of the vertex of the graph of (1) are
$$\left( \frac{\mathbf{A}}{a}, -\frac{\mathbf{B}}{a} - 4a \right).$$
(2) Among the following choices, the quadratic function whose graph is $G$ is $\square$ C. (0) $y = ax^2 + 4x + 4a$
(1) $y = ax^2 + 4x - 4a$
(2) $y = ax^2 - 4x + 4a$
(3) $y = -ax^2 + 4x + 4a$
(4) $y = -ax^2 - 4x + 4a$
(5) $y = -ax^2 - 4x - 4a$
(3) The curve $G$ intersects the graph of the quadratic function (1) at the two points
$$(\mathrm{DE},\ \mathrm{F}) \text{ and } (\mathrm{G},\ \mathrm{HI}).$$
(4) Let $a = 2$. Then over the interval $\mathrm{DE} \leqq x \leqq \square\mathrm{G}$, the maximum and the minimum values of the quadratic function whose graph is $G$ are JK and LM, respectively.