Let's analyze the quadratic functions given and determine their vertices. The general form of a vertex in a quadratic equation of the form $y = a(x - h)^2 + k$ is $(h, k)$, where $h$ and $k$ are the coordinates of the vertex.

Here are the given functions:

1. $y = 2(x - 1)^2 + 4$

   • The vertex form is clear: $h = 1$, $k = 4$.
   • So the vertex is $(1, 4)$, which corresponds to F.

2. $y = (x + 3)^2 + 2$

   • This can be written as $y = (x - (-3))^2 + 2$, so $h = -3$, $k = 2$.
   • The vertex is $(-3, 2)$, which corresponds to A.

3. $y = (x - 3)^2 - 2$

   • The vertex form gives $h = 3$, $k = -2$.
   • The vertex is $(3, -2)$, which corresponds to E.

4. $y = 2x^2 + 4x - 2$

   • First, we need to convert this to vertex form. Let's complete the square: $y = 2(x^2 + 2x) - 2$ $y = 2((x + 1)^2 - 1) - 2$ $y = 2(x + 1)^2 - 2 - 2 = 2(x + 1)^2 - 4$ The vertex form is $y = 2(x + 1)^2 - 4$, so $h = -1$, $k = -4$.
     • The vertex is $(-1, -4)$, which corresponds to D.

5. $y = (x - 3)^2 + 2$

   • The vertex form gives $h = 3$, $k = 2$.
   • The vertex is $(3, 2)$, which corresponds to C.

Final Answers:

• $y = 2(x - 1)^2 + 4$: F
• $y = (x + 3)^2 + 2$: A
• $y = (x - 3)^2 - 2$: E
• $y = 2x^2 + 4x - 2$: D
• $y = (x - 3)^2 + 2$: C

Let me know if you need further details or have any questions!

Here are some follow-up questions:

1. What does the "a" value in the vertex form equation $y = a(x - h)^2 + k$ represent?
2. How can you determine if the parabola opens upwards or downwards?
3. How would you find the vertex if the quadratic equation is in standard form $ax^2 + bx + c$?
4. What role does the vertex play in identifying the axis of symmetry of a parabola?
5. How does completing the square help in converting from standard form to vertex form?

Tip: When converting a quadratic equation to vertex form, completing the square is a reliable technique to easily identify the vertex.