Learn the following:

y = A(x - H)² + K 

is the equation of a curve called a parabola, that either looks like this

 or like this: 

If it looks like this , we say it OPENS UPWARD

If it looks like this , we say it OPENS DOWNWARD
 

The turning point (called the "vertex") is at the bottom of the graph
if it looks like this  

The turning point or vertex is at the top of the graph if it looks 
like this .

The vertex is the point (H,K).

If A, the coefficient of the parentheses, is positive, the graph will
look like this: 

If A, the coefficient of the parentheses, is negative, the graph will
look like this: .

Two points on each side of the vertex which the parabola goes through
are (H-1,K+A) and (H+1,K+A)



Now let's compare your equation:

y = -(x + 2)² + 2

with this one:

 y =  A(x - H)² + K

Put a 1 between the negative sign and the parentheses:

 y = -1(x + 2)² + 2

You can see that A = -1, 

You can see that -H = +2, and so H = -2

You can see that K = 2

Therefore the vertex (turning point) is (H,K) = (-2,2)

Since A is a negative number the parabola open downward and so it will
look like this: 

Two other points are (H-1,K+A) and (H+1,K+A).
These are 

(H-1,K+A) = (-2-1,2-1) = (-3,1) 

and 

(H+1,K+A = (-2+1,2-1) = (-1,1)

So we plot those three points, the vertex (-2,2), and the two points
(-3,1) and (-1,1), like this::



Then we can sketch in the graph:



It has the axis of symmetry, which is a vertical line through the
vertex, which has the equation 

x = H

 which in your case is the equation 

x = -2, 

and is the green vertical line below that goes through
x = -2 on the x-axis:

 

Edwin