I'll help you solve the problems from the image you've uploaded. Let me first analyze the content and extract the necessary information to proceed with solving the questions.

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.