You’ve shared a list of quadratic equations in the form of vertex form, which are equations of parabolas. Each of these equations represents a parabola, and the parameters in each equation provide valuable information about its shape, position, and direction.

General form:

The general form for a parabola in vertex form is: $y = a(x - h)^2 + k$ Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ determines the direction (whether the parabola opens upwards or downwards) and how "wide" or "narrow" the parabola is.

Detailed information from each equation:

1. $y = -\frac{1}{2}(x - 2)^2 + 4$

   • Vertex: $(2, 4)$
   • Opens downwards (since $a = -\frac{1}{2}$).
   • "Wider" parabola (since $|a| = \frac{1}{2}$).

2. $y = \frac{1}{2}(x + 3)^2 - 1$

   • Vertex: $(-3, -1)$
   • Opens upwards (since $a = \frac{1}{2}$).
   • "Wider" parabola (since $|a| = \frac{1}{2}$).

3. $y = -\frac{1}{4}(x - 1)^2 + 3$

   • Vertex: $(1, 3)$
   • Opens downwards (since $a = -\frac{1}{4}$).
   • "Wider" parabola (since $|a| = \frac{1}{4}$).

4. $y = \frac{1}{2}(x - 2)^2 - 2$

   • Vertex: $(2, -2)$
   • Opens upwards (since $a = \frac{1}{2}$).
   • "Wider" parabola (since $|a| = \frac{1}{2}$).

5. $y = -\frac{1}{3}(x + 4)^2 + 5$

   • Vertex: $(-4, 5)$
   • Opens downwards (since $a = -\frac{1}{3}$).
   • "Wider" parabola (since $|a| = \frac{1}{3}$).

6. $y = \frac{1}{2}(x - 1)^2 - 3$

   • Vertex: $(1, -3)$
   • Opens upwards (since $a = \frac{1}{2}$).
   • "Wider" parabola (since $|a| = \frac{1}{2}$).

7. $y = -\frac{2}{3}(x + 2)^2 + 4$

   • Vertex: $(-2, 4)$
   • Opens downwards (since $a = -\frac{2}{3}$).
   • "Narrower" parabola (since $|a| = \frac{2}{3}$).

8. $y = \frac{1}{3}(x - 3)^2 - 1$

   • Vertex: $(3, -1)$
   • Opens upwards (since $a = \frac{1}{3}$).
   • "Wider" parabola (since $|a| = \frac{1}{3}$).

9. $y = -\frac{1}{4}(x + 1)^2 + 2$

   • Vertex: $(-1, 2)$
   • Opens downwards (since $a = -\frac{1}{4}$).
   • "Wider" parabola (since $|a| = \frac{1}{4}$).

10. $y = \frac{2}{3}(x - 2)^2 - 4$

    • Vertex: $(2, -4)$
    • Opens upwards (since $a = \frac{2}{3}$).
    • "Narrower" parabola (since $|a| = \frac{2}{3}$).

11. $y = -\frac{1}{2}(x + 5)^2 + 6$

    • Vertex: $(-5, 6)$
    • Opens downwards (since $a = -\frac{1}{2}$).
    • "Wider" parabola (since $|a| = \frac{1}{2}$).

12. $y = \frac{1}{4}(x - 3)^2 - 2$

    • Vertex: $(3, -2)$
    • Opens upwards (since $a = \frac{1}{4}$).
    • "Wider" parabola (since $|a| = \frac{1}{4}$).

Summary:

• These equations represent parabolas with varying vertex positions, directions (up or down), and widths (narrow or wide).
• The vertex of each parabola is given by the values $(h, k)$.
• The sign of $a$ determines the direction: negative $a$ means the parabola opens downward, while positive $a$ means it opens upward.
• The magnitude of $a$ controls the width of the parabola: smaller $|a|$ values make the parabola wider, while larger $|a|$ values make it narrower.

Would you like to explore how these parabolas relate graphically, or need further explanations on any of the specific equations?