https://file.aiquickdraw.com/mathbot/file/1724041186532w10hx3my.jpg

Vertex:  (0, 1)
Focus:  (0, 2.5) 
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

Given the vertex at \((0, 1)\) and the focus at \((0, 2.5)\), we can write the equation of the parabola as follows:

- The parabola opens upwards because the focus is above the vertex.
- The distance between the vertex and the focus is \(2.5 - 1 = 1.5\) units. This distance represents \(\frac{1}{4a}\), so \(a = \frac{1}{4 \times 1.5} = \frac{1}{6}\).

Using the vertex form of the equation \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex:

\[ y = \frac{1}{6}(x - 0)^2 + 1 \]

Simplifying this:

\[ y = \frac{1}{6}x^2 + 1 \]

Therefore, the sentence should read:

**"An equation of the parabola is \( y = \frac{1}{6}x^2 + 1 \)."**

### Brief Explanation:
The equation was derived using the vertex form \(y = a(x-h)^2 + k\), with the vertex at \((0, 1)\) and the value of \(a\) calculated based on the distance between the vertex and the focus, which is \(1.5\) units.

Would you like any more details or have any questions?

Here are some related questions:
1. How does the distance between the vertex and the focus affect the shape of the parabola?
2. How can you determine whether a parabola opens upwards or downwards?
3. What is the significance of the vertex in determining the equation of a parabola?
4. How would the equation change if the parabola opened downwards?
5. How do you find the directrix of a parabola given the vertex and focus?

**Tip:** Remember that the distance between the vertex and the focus helps determine the steepness of the parabola. The larger this distance, the narrower the parabola.

Understand how to find the equation of a parabola given its vertex and focus. This problem involves calculating parameters using the vertex form of a parabola and understanding the relationship between the vertex and focus. Suitable for Grades 9-12.

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