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 $$y = \mathbf { L } x ^ { 2 } + \mathbf { M } .$$
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
$$y = \mathbf { L } x ^ { 2 } + \mathbf { M } .$$