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write an equation of the parabola shown. answer using this exact sentence, "An equation of the parabola is _________". Briefly explain how you got that answer

To write the equation of the parabola shown in the image, let's analyze the details:

- **Vertex**: The vertex of the parabola is at the point \((0, 2)\).
- **Focus**: The focus of the parabola is at the point \((0, 3)\).

Since the parabola opens upwards and its vertex form equation is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex, the equation can be determined by the distance between the vertex and the focus.

The distance between the vertex and the focus is 1 unit (from \(y = 2\) to \(y = 3\)). This distance is equal to \( \frac{1}{4a} \), so \( a = \frac{1}{4 \times 1} = \frac{1}{4} \).

Now, substituting the values into the vertex form equation:

\[ y = \frac{1}{4}(x - 0)^2 + 2 \]

Simplifying this:

\[ y = \frac{1}{4}x^2 + 2 \]

Thus, the sentence should read:

**"An equation of the parabola is \( y = \frac{1}{4}x^2 + 2 \)."**

### Brief Explanation:
The equation was derived using the vertex form \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex \((0, 2)\). The value of \(a\) was determined by the distance between the vertex and the focus, which is \( \frac{1}{4} \), given the focus is 1 unit above the vertex.

Would you like any more details or have any questions?

Here are some related questions:
1. How do you find the value of \(a\) in a parabola?
2. What is the significance of the focus in a parabola?
3. How does the direction a parabola opens affect its equation?
4. How can you find the equation of a parabola if given the directrix instead of the focus?
5. What role does the vertex play in the equation of a parabola?

**Tip:** When writing the equation of a parabola, always ensure you correctly identify the vertex and the direction in which the parabola opens.

Learn how to write the equation of a parabola given its vertex and focus points. This problem involves using the vertex form of a parabola and calculating the distance between key points. Suitable for high school students studying algebra and geometry.

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