Given:
The graph of [tex]y = ax^2+c[/tex] contains the point (2, 14).
To find:
The point which lies on the graph of [tex]y = a(x-2)^2+c[/tex].
Solution:
Consider the given equations are
[tex]y_1 = ax^2+c[/tex] ...(i)
[tex]y_2 = a(x-2)^2+c[/tex] ...(ii)
The translation is defined as
[tex]g(x)=f(x+a)[/tex]
where, a is horizontal shift.
If a>0, then the graph shifts a units left and if a<0, then the graph shifts a units right.
From equation (i) and (ii), it is clear that a=-2. So, graph of [tex]y = ax^2+c[/tex] shifts 2 units right to get the graph of [tex]y = a(x-2)^2+c[/tex].
It means each point on [tex]y = ax^2+c[/tex] shifts 2 units right.
[tex](x,y)\to (x+2,y)[/tex]
(2, 14) lies on [tex]y = ax^2+c[/tex].
[tex](2,14)\to (2+2,14)[/tex]
[tex](2,14)\to (4,14)[/tex]
Therefore, (4,14) must be lies on [tex]y = a(x-2)^2+c[/tex].
