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Please give me a moment.The image contains a worksheet with several problems that involve graphing and interpreting quadratic functions. I will break down the specific problems visible and provide solutions accordingly.

### 1. **Quadratic Graph Analysis**
   The first graph shows a parabola, and you're asked to identify the vertex and axis of symmetry.
   - **Vertex:** The vertex is the point where the parabola changes direction. In this graph, the vertex is at \( (2, -3) \).
   - **Axis of Symmetry:** The axis of symmetry is a vertical line that passes through the vertex. Since the vertex is at \( x = 2 \), the equation of the axis of symmetry is \( x = 2 \).

### 2. **Graphing Based on Vertex and Axis of Symmetry**
   Another question asks you to graph a quadratic function by using the vertex form of the equation. The given vertex is \( (2, -3) \), and the axis of symmetry is \( x = 2 \).

   - **Using the vertex form equation:** The vertex form of a quadratic function is:
   \[
   y = a(x - h)^2 + k
   \]
   where \( (h, k) \) is the vertex. For this case, since the vertex is \( (2, -3) \), the equation is:
   \[
   y = a(x - 2)^2 - 3
   \]
   You would need more information (like another point) to determine the value of \( a \).

### 3. **Another Graph and Equation Interpretation**
   Another graph shows a quadratic function with a vertex at \( (-2, 5) \). This question likely asks for the vertex and axis of symmetry:
   - **Vertex:** \( (-2, 5) \)
   - **Axis of Symmetry:** \( x = -2 \)

### 4. **Completing the Square**
   There is also a problem asking you to complete the square for the quadratic equation \( x^2 + 4x + y = 6 \). After rearranging terms, you need to rewrite the equation in vertex form:
   \[
   y = x^2 + 4x - 6
   \]
   Completing the square for the quadratic expression \( x^2 + 4x \):
   \[
   x^2 + 4x = (x + 2)^2 - 4
   \]
   Substituting this back, we get the vertex form:
   \[
   y = (x + 2)^2 - 10
   \]
   The vertex of this parabola is \( (-2, -10) \).

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Would you like further details on any of these problems or any additional questions?

Here are 5 related questions:
1. How do you determine the direction of a parabola?
2. What role does the coefficient \( a \) play in the vertex form of a quadratic function?
3. How do you find the focus and directrix of a parabola?
4. How can you identify the roots of a quadratic equation graphically?
5. How do transformations affect the graph of a quadratic function?

**Tip:** Completing the square is a powerful technique for converting a quadratic equation into vertex form, which is very useful for graphing.

Explore solutions to problems involving graphing and interpreting quadratic functions. Learn how to identify vertices, axis of symmetry, and use vertex form equations. Suitable for high school students studying algebra and quadratic equations.

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