Consider the quadratic function $$y = 3 x ^ { 2 } - 6 .$$ (1) Suppose that the graph obtained by a parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ passes through the two points $( 1,5 )$ and $( 4,14 )$. The quadratic function of this graph is $$y = \mathbf { A } x ^ { 2 } - \mathbf { BC } x + \mathbf { D E } .$$ This graph is the parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ by $\mathbf{F}$ in the $x$-direction and by $\mathbf { G }$ in the $y$-direction. (2) The quadratic function having the graph which is symmetric to the graph of $y = 3 x ^ { 2 } - 6$ with respect to the straight line $y = c$ is $$y = - \mathbf { H } x ^ { 2 } + \mathbf { I } c + \mathbf { J } .$$ When the graphs of the two quadratic functions (1) and (2) have just one common point, it follows that $c = \mathbf { K }$, and the coordinates of the common point are ( $\mathbf { L } , \mathbf { M }$ ).
Consider the quadratic function
$$y = 3 x ^ { 2 } - 6 .$$
(1) Suppose that the graph obtained by a parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ passes through the two points $( 1,5 )$ and $( 4,14 )$. The quadratic function of this graph is
$$y = \mathbf { A } x ^ { 2 } - \mathbf { BC } x + \mathbf { D E } .$$
This graph is the parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ by $\mathbf{F}$ in the $x$-direction and by $\mathbf { G }$ in the $y$-direction.
(2) The quadratic function having the graph which is symmetric to the graph of $y = 3 x ^ { 2 } - 6$ with respect to the straight line $y = c$ is
$$y = - \mathbf { H } x ^ { 2 } + \mathbf { I } c + \mathbf { J } .$$
When the graphs of the two quadratic functions (1) and (2) have just one common point, it follows that $c = \mathbf { K }$, and the coordinates of the common point are ( $\mathbf { L } , \mathbf { M }$ ).