Let $f(x)=x^2$ be the parent function for the transformation $g(x)=(2x+12)^2 -1$. Follow the two points $(0,0)$ and $(2,4)$ from the parent function through the transformation.
Let's simplify $g(x)$ first. So, $g(x)=(2(x+6))^2 -1$. This form tells me the transformations of the graph.
I know that the $x$-values from the parent function, call them $h$, will be transformed by $\frac{h}{2}-6$. Also, I know that the $y$-values from the parent function, call them $k$, will be transformed to $y=k-1$.
So, the points $(0,0)$ and $(2,4)$ are mapped to $(-6,-1)$ and $(-5,3)$, respectively.
Now, if I simplify $g(x)$ to $g(x)=4(x+6)^2 -1$, then parent points $(h,k)$ transform to $(h-6, 4k-1)$. So, the points $(0,0)$ and $(2,4)$ are mapped to $(-6,-1)$ and $(-4,15)$. The last point is different from what I original calculated. That is, the point $(2,4)$ was mapped to different locations on $g(z)$.
These functions are the same but simplified differently. However, the parent points are mapped to different locations on the transformed function. Why does this happen? I'm assuming that both will eventually map the entire graph of the transformed function, but I thought that the parent points would be mapped to the same set of points.