Find the equation of a parabola with vertex at (2, 3) and focus at (2,  5).

To find the equation of a parabola with a given vertex and focus, we can use the following information:

- The vertex is \((h, k)\).
- The focus is \((h, k + p)\) if the parabola opens vertically.

Given:
- Vertex \((h, k) = (2, 3)\)
- Focus \((h, k + p) = (2, 5)\)

### Step 1: Determine the Value of \(p\)
The distance between the vertex and the focus is \(p\). Since the focus is at \((2, 5)\) and the vertex is at \((2, 3)\), we can calculate \(p\) as:
\[
p = 5 - 3 = 2
\]

### Step 2: Determine the Orientation of the Parabola
Since the focus is above the vertex, the parabola opens upwards.

### Step 3: Use the Standard Form of the Parabola
The standard form of the equation for a parabola that opens upwards or downwards is:
\[
(y - k) = \frac{1}{4p}(x - h)^2
\]

Substituting the values \(h = 2\), \(k = 3\), and \(p = 2\):
\[
y - 3 = \frac{1}{4 \times 2}(x - 2)^2
\]

Simplify the equation:
\[
y - 3 = \frac{1}{8}(x - 2)^2
\]

### Step 4: Write the Final Equation
The final equation of the parabola is:
\[
y = \frac{1}{8}(x - 2)^2 + 3
\]

This is the equation of the parabola with vertex \((2, 3)\) and focus \((2, 5)\).

Would you like any further details or have any questions?

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Here are five related questions:

1. How do you find the equation of a parabola given the vertex and directrix?
2. What is the significance of the directrix in relation to the focus?
3. How do you determine if a parabola opens left, right, up, or down?
4. How can you find the focus of a parabola if you only know the vertex and the equation?
5. What changes in the equation of the parabola if the focus is below the vertex?

**Tip:** The distance between any point on a parabola and its focus is equal to the distance between that point and the directrix.

Learn how to find the equation of a parabola with vertex at (2, 3) and focus at (2, 5). This problem involves concepts of analytic geometry and parabolas. Suitable for high school students studying conic sections.

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