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give me the vertex equation of this graph show all work

To find the equation of the parabola and its vertex from the provided graph, I will follow these steps:

### Observing the graph:
1. The graph is a parabola opening upwards, indicating a positive quadratic term.
2. The vertex appears to be at the point \( (0, 5) \), based on the symmetry and minimum value.

### General Vertex Form of a Parabola:
The equation of a parabola in vertex form is:

\[
y = a(x - h)^2 + k
\]

Where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) determines the width and direction of the parabola.

From the graph, we can observe:
- The vertex is \( (0, 5) \), so \( h = 0 \) and \( k = 5 \).

Thus, the equation simplifies to:

\[
y = a(x - 0)^2 + 5 \quad \text{or} \quad y = ax^2 + 5
\]

### Finding \( a \):
To determine the value of \( a \), we need another point on the parabola. From the graph, one easily identifiable point is \( (1, 6) \), as it seems to lie on the curve.

Substituting \( x = 1 \) and \( y = 6 \) into the equation:

\[
6 = a(1)^2 + 5
\]

Solving for \( a \):

\[
6 = a + 5
\]
\[
a = 1
\]

### Final Equation:
Thus, the equation of the parabola is:

\[
y = x^2 + 5
\]

This is the equation of the parabola shown in the graph.

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Would you like further details on the steps? Here are some related questions:

1. What is the general form of a quadratic equation, and how does it relate to the vertex form?
2. How do you determine whether a parabola opens upwards or downwards?
3. How can you determine the vertex from a graph without the equation?
4. What effect does changing the value of \( a \) have on the graph of the parabola?
5. Can this equation be converted into standard form, and how?

**Tip:** The coefficient \( a \) affects the "width" or "narrowness" of the parabola—higher absolute values make it narrower, while lower values make it wider.

This page provides a detailed step-by-step solution for finding the vertex equation of a parabola from a graph. The problem involves key algebraic concepts, including the vertex form of a quadratic equation and identifying the vertex point on a graph. Suitable for students in Grades 9-12.

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