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

Jun 18, 2015

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

$y + 3 = {\left(x - 2\right)}^{2}$

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 $y = {\left(x - 2\right)}^{2}$, after we square, we are done, that is the $y$ value.

In $y = {\left(x - 2\right)}^{2} + 3$, 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 $\left(0 , 0\right)$

The vertex of $y = {\left(x - 2\right)}^{2} - 3$ is the point $\left(2 , - 3\right)$.