The table seems to involve analyzing quadratic equations and their characteristics. Let me help you interpret and complete the missing elements of the table based on the given equations and other details. I will extract the necessary information step by step:

Quadratic Equations in the Table

Here are the equations visible in the table:

1. $y = (x + 2)^2$
2. $y = 3x^2 - 6$
3. $y = -0.5(x + 3)^2 + 5$
4. $y = 2(x - 1)^2 - 4$

Characteristics to Analyze

For each equation, you need to determine the following:

1. Direction of Opening: Depends on the coefficient of the $x^2$ term.
2. Vertex: Use the vertex form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
3. Axis of Symmetry: The vertical line passing through the vertex, $x = h$.
4. Optimal Value: The $y$-coordinate of the vertex (maximum if parabola opens down, minimum if it opens up).
5. Step Pattern: Based on the coefficient of $x^2$.
6. Translations (Shifts): Horizontal and vertical shifts from the standard $y = x^2$.
7. Vertical Compression/Stretch/Reflection: Based on the coefficient $a$ in the equation.

Let me analyze each equation and fill in the details.

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

• Direction of Opening: Up (coefficient of $x^2$ is positive).
• Vertex: $(-2, 0)$ (from $(x + 2)^2$).
• Axis of Symmetry: $x = -2$.
• Optimal Value: $0$ (minimum at vertex).
• Step Pattern: Standard (1, 3, 5...).
• Translations (Shifts): Left 2, no vertical shift.
• Vertical Compression/Stretch/Reflection: None.

2. $y = 3x^2 - 6$

• Direction of Opening: Up (coefficient of $x^2$ is positive).
• Vertex: $(0, -6)$ (from standard form, vertex at $(0, c)$).
• Axis of Symmetry: $x = 0$.
• Optimal Value: $-6$ (minimum at vertex).
• Step Pattern: Stretched by 3 (3, 9, 15...).
• Translations (Shifts): Down 6, no horizontal shift.
• Vertical Compression/Stretch/Reflection: Stretched by a factor of 3.

3. $y = -0.5(x + 3)^2 + 5$

• Direction of Opening: Down (coefficient of $x^2$ is negative).
• Vertex: $(-3, 5)$ (from $(x + 3)^2$ and $+5$).
• Axis of Symmetry: $x = -3$.
• Optimal Value: $5$ (maximum at vertex).
• Step Pattern: Compressed by 0.5 and inverted (-0.5, -1.5, -2.5...).
• Translations (Shifts): Left 3, up 5.
• Vertical Compression/Stretch/Reflection: Compressed by a factor of 0.5 and reflected.

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

• Direction of Opening: Up (coefficient of $x^2$ is positive).
• Vertex: $(1, -4)$ (from $(x - 1)^2$ and $-4$).
• Axis of Symmetry: $x = 1$.
• Optimal Value: $-4$ (minimum at vertex).
• Step Pattern: Stretched by 2 (2, 6, 10...).
• Translations (Shifts): Right 1, down 4.
• Vertical Compression/Stretch/Reflection: Stretched by a factor of 2.

Let me know if you want me to organize these into a clean table or clarify further.

Relative Questions:

1. How do you determine the direction of opening for quadratic equations?
2. What is the significance of the vertex in a parabola, and how do you find it?
3. How does the coefficient of $x^2$ affect the step pattern and graph shape?
4. Can you explain how translations affect the standard $y = x^2$ graph?
5. What is the relationship between the axis of symmetry and the vertex?

Tip:

Always rewrite quadratic equations in vertex form $y = a(x - h)^2 + k$ to easily identify the vertex and translations.