Here the list of the transformations applied to the parent function, and their effect:
- [tex] x^2 \to (2x)^2=4x^2 [/tex]. This causes a horizontal compression of factor 4.
- [tex] (2x)^2 \to (2x+6)^2= 4(x+3)^2 [/tex]. This causes a horizontal translation, 3 units to the left.
- [tex] (2x+6)^2 \to -(2x+6)^2 [/tex]. This causes a reflection about the x axis.
- [tex]-(2x+6)^2 \to -(2x+6)^2+3 [/tex]. This causes a vertical translation, 3 units up
So, if you start from the graph of [tex] y=x^2 [/tex], you have to compress and translate it horizontally, flip it with respect to the x axis, and finally move it up 3 units.
