About finding the points that result. You can apply the transformations to the points as you go or you can find the intercepts and notable points after with a little algebra. e.g. for 4e. to find the x-intercepts you have

y=2-|x^{2}-2|

x-intercepts, y=0

2-|x^{2}-2|=0

|x^{2}-2|=2

now the mod will make a negative value inside it positive and leave a positive value as positive.

i.e. x^{2}-2=2 (when x^{2}-2>0) and -(x^{2}-2)=2 (when x^{2}-2<0)

x^{2}=4 or x^{2}=0

x=+-2 or x=0

i.e. x-intercepts at (-2,0), (2,0) and (0,0)

Do the same for the y-intercept and use the transformations to find the turning point.

EDIT: e.g. original t.p at (0,-2)

Apply the mod, all the negative y-values will become positive as it is flipped in the x-axis.

(0,-2)->(0,2)

Then apply the whole reflection of the graph in the x-axis.

(0,2)->(0,-2)

Then apply the shift of two units up.

(0,-2)->(0,0)