How do you identify transformations in parent functions given #y = 2(x-3)^2#?

Look at the a-value and h-value (and the k-value): the a-value transforms the parabola. The h and k-value translates the parabola.

In this case, the transformed parabola has a vertical stretch by a factor of 2 and is translated 3 units to the right. The parent function if y=x^2, which looks like this:

graph{x^2 [-10, 10, -5, 5]}

The transformed function, y=2(x-3)^2 is a lot more simpler to determine its transformation because it is given in vertex form.

There are two main things being done to the parabola.

The a-value - having a value greater than 1 as the a-value indicates a vertical stretch by a factor of whatever was used. In this case, 2.

The h-value translates the parabola to the left or right. It is determined by isolating the x-value in the bracket (AND BRACKET ONLY). In this case, the parabola is moved 3 units to the right.

It looks like this:

graph{2(x-3)^2 [-10, 10, -5, 5]}

Hope this helps