# How do you sketch the graph of #y=-(x+2)^2+2 and describe the transformation?

Dec 17, 2017

The transformations are: shift two units to the left (horizontal shift), reflect over the $x$-axis, shift two units up (vertical shift)

#### Explanation:

Begin with the graph of the parent function: $y = {x}^{2}$
graph{x^2 [-10, 10, -5, 5]}
Now we want to deal with each of the transformations one at a time.

Looking at $y = - {\left(x + 2\right)}^{2} + 2$ the first transformation is to shift the graph 2 units to the left because of the $x + 2$ in parentheses.

That gives us this graph:
graph{(x+2)^2 [-10, 10, -5, 5]}
The vertex moved from $\left(0 , 0\right)$ to $\left(- 2 , 0\right)$ with that transformation.

Looking back at our function, $y = - {\left(x + 2\right)}^{2} + 2$, the next transformation to deal with is that negative. That will cause our graph to reflect over the $x$-axis. The vertex doesn't change with this transformation.

Now the graph looks like:
graph{-(x+2)^2 [-10, 10, -5, 5]}

Finally we want to deal with the +2 at the end of the function. That will take the entire graph and shift it two units up (vertically). This changes the vertex to $\left(- 2 , 2\right)$.

Here's the final graph:

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

So the transformations are: shift two units to the left (horizontal shift), reflect over the $x$-axis, shift two units up (vertical shift).