Question

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

Answer 1

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=-(x+2)^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 `(0,0)` to `(-2,0)` with that transformation.

Looking back at our function, `y=-(x+2)^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 `(-2,2)`.

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).