Question

How to graph a parabola `y=(x-2)^2-3`?

Answer 1

The graph has the same shape as y = x^2, but there are some shifts

Explanation:

Replacing `x` with `x-2` makes `x=2` act in the new equation just like `x=0` did in the old one. (That is where I would find `0^2`.)
That shifts the graph `2` to the right.

Compare `y = x^2` and

`y+3 = (x-2)^2`

Replacing `x` with `x+2` moves the graph `2` in the positive `x` direction (2 to the right.)

What do we expect to happen when we replace `y` with `y+3`?
If you said "move the graph `3` in the negative `y` direction (3 downward)", then you are right!

A different way of thinking about it: After we find the square, what do we do?

In `y = x^2` we're done, that is the `y` value.

In `y = (x-2)^2`, after we square, we are done, that is the `y` value.

In `y = (x-2)^2 +3`, after we square, we still need to subtract 3 from the number, that moves us down 3.

The vertex of `y=x^2` is the point `(0,0)`

The vertex of `y = (x-2)^2-3` is the point `(2,-3)`.