Also these 2 :) determining the vertex and shifts/translations

The answer to your question is below

Step-by-step explanation:

A) f(x) = x² + 4x + 1

Find the vertex

       y = x² + 4x + 1

-Subtract 1 in both sides

      y - 1 = x² + 4x

-Complete the perfect trinomial square in the right side

     y - 1 + (2)² = x² + 4x + (2)²

-Simplify

     y - 1 + 4 = (x + 2)²

     y + 3 = (x + 2)²

-Find the vertex

      V = (-2, -3)

B)

   f(x) = x²

   g(x) = (x - 4)² - 2      See the picture below

Shifted right           4 units

Shifted down         2 units