Find the general equation of the parabola with vertex at (3, 6) and focus at (3, 4). Sketch the graph.

Solution:

Let's solve the problem where the parabola has a vertex at \((3, 6)\) and a focus at \((3, 4)\), and then sketch the graph.

### Solution:

1. **Identify the vertex and focus:**
   - **Vertex:** \((h, k) = (3, 6)\)
   - **Focus:** \((h, k - p) = (3, 4)\)

2. **Determine the orientation of the parabola:**
   - Since the focus \((3, 4)\) is below the vertex \((3, 6)\), the parabola opens **downward**.

3. **Calculate the distance \(p\) between the vertex and the focus:**
   - The distance \(p\) is the difference in the \(y\)-coordinates of the vertex and the focus:
     \[
     p = k - y_{\text{focus}} = 6 - 4 = 2
     \]
   - So, \(p = 2\).

4. **Write the standard form equation of the parabola:**
   - For a parabola that opens vertically (upward or downward), the general equation is:
     \[
     (y - k) = \frac{1}{4p}(x - h)^2
     \]
   - Since the parabola opens downward, we have \(p = -2\). Substituting the values \(h = 3\), \(k = 6\), and \(p = 2\):
     \[
     (y - 6) = \frac{1}{4(-2)}(x - 3)^2
     \]
     \[
     (y - 6) = -\frac{1}{8}(x - 3)^2
     \]
   - The general equation of the parabola is:
     \[
     y = -\frac{1}{8}(x - 3)^2 + 6
     \]

### Graph of the Parabola:

1. **Plot the vertex** at \((3, 6)\).
2. **Plot the focus** at \((3, 4)\), which is 2 units below the vertex.
3. **Draw the axis of symmetry,** which is the vertical line \(x = 3\).
4. **Draw the directrix,** which is the horizontal line \(y = 8\) (since the directrix is \(p\) units above the vertex).
5. **Sketch the parabola:** The parabola will open downward, with the vertex at the highest point and curving downward through the focus.

Would you like to see the sketch, or do you need further details?

Here are some related questions to consider:

1. How do you determine the direction of a parabola based on its focus and vertex?
2. What role does the directrix play in defining the parabola?
3. How would the equation change if the parabola opened upward instead of downward?
4. Can you derive the equation of a parabola given only its directrix and a point on the parabola?
5. What are the key features (vertex, focus, directrix, axis of symmetry) that uniquely define a parabola?

**Tip:** The vertex form of a parabola’s equation is useful because it clearly shows the vertex’s location and helps in identifying the direction in which the parabola opens.

Learn how to find the general equation of a parabola with vertex at (3, 6) and focus at (3, 4). This problem involves understanding the relationship between the vertex, focus, and the standard form of the parabola equation. Suitable for high school students.

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