Which rule describes the function in the graph below

We know that
a) for the interval (-∞, -2] the graph is a line
find the equation of the line
let
A(-4,3)
B(-2,2)
slope m=(2-3)/(-2+4)> m=-1/2
with m=-1/2 and B(-2,2) find the equation of the line
y-y1=m*(x-x1)> y-2=(-1/2)*(x+2)> y=(-1/2)x-1+2
y=-0.5x+1

for the interval (-∞, -2]
the rule is
y=-0.5x+1

Part b) for the interval [-2,0] the graph is a vertical parabola open down
the vertex is the point (0,6)

(y-k)=a(x-h)²
for x=-2  y=2
(2-6)=a(-2-0)²> -4=4a> a=-1
the equation is 
y-6=a(x)²> y=-x²+6
y=-x²+6

for the interval [-2, 0]
the rule is
y=-x²+6

Part c) for the interval [0,4] the graph is a line
y=6
for the interval [0,4]
the rule is
y=6

Part d) for the interval [4, ∞) the graph is a line
find the equation of the line
let
A(4,6)
B(6,8)
slope m=(8-6)/(6-4)> m=1
with m=1 and B(6,8) find the equation of the line
y-y1=m*(x-x1)> y-8=(1)*(x-6)> y=(x-6)+8
y=x+2

for the interval [4, ∞)
the rule is
y=x+2