Greetings from Brasil...
The vertex is coordinates are given by:
V = (-b/2a ; -Δ/4a)
P(X) = (X + 3)² + 2 (notable products)
(X + 3)² = X² + 2.X.3 + 3² = X² + 6X + 9 so
P(X) = X² + 6X + 9 + 2
P(X) = X² + 6X + 11
- b/2a = - 6/2.1 = - 3
-Δ/4a
Δ = b² - 4.a.c
Δ = 6² - 4.1.11
Δ = 36 - 44
Δ = - 8
- Δ/4a = - (- 8)/4.1 = 8/4 = 2
V = (-3; 2)
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translation = f(x) + c ⇒ P(X) + c
P(X) + c = (X + 3)² + 2 + c
P(X) + 3 = (X + 3)² + 2 + 3
Q(X) = (X + 3)² + 5
The translation was only upwards, so the coordinate X does not change (only Y), so
V = (-3; 2 + 3)
V = (-3; 5)
