Graph the parabola of the focus=(2,2) and Dentrix=x=-8

To graph the parabola given the focus \((2, 2)\) and the directrix \(x = -8\), we need to determine the equation of the parabola. 

1. **Determine the vertex**: The vertex is the midpoint between the focus and the directrix. 

   - The focus is \((2, 2)\).
   - The directrix is \(x = -8\).

   The midpoint between the focus \((x_f, y_f)\) and the directrix \(x = -8\) is given by:

   \[
   \left( \frac{x_f + x_d}{2}, y_f \right) = \left( \frac{2 + (-8)}{2}, 2 \right) = (-3, 2)
   \]

   So, the vertex of the parabola is \((-3, 2)\).

2. **Determine the orientation of the parabola**: Since the directrix is vertical (\(x = -8\)), the parabola opens horizontally. The directrix is to the left of the vertex, so the parabola opens to the right.

3. **Determine the distance from the vertex to the focus**: 

   The distance from the vertex \((-3, 2)\) to the focus \((2, 2)\) is 5 units. This distance is the value of \(p\), where \(p\) is the distance from the vertex to the focus (or vertex to the directrix).

   Since the parabola opens to the right, \(p\) is positive.

4. **Write the equation of the parabola**: 

   For a parabola that opens horizontally (to the right), the standard form of the equation is:

   \[
   (y - k)^2 = 4p(x - h)
   \]

   Here, \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus.

   Substituting the vertex \((-3, 2)\) and \(p = 5\):

   \[
   (y - 2)^2 = 4 \cdot 5 \cdot (x + 3)
   \]

   Simplifying:

   \[
   (y - 2)^2 = 20(x + 3)
   \]

So, the equation of the parabola is:

\[
(y - 2)^2 = 20(x + 3)
\]

### Graph of the Parabola

To graph the parabola, you can follow these steps:

1. **Plot the vertex** at \((-3, 2)\).
2. **Plot the focus** at \((2, 2)\).
3. **Draw the directrix** as the vertical line \(x = -8\).
4. **Sketch the parabola** opening to the right, ensuring that it passes through the vertex and curves around the focus.

Would you like a detailed graph of this parabola? If so, please upload a photo or ask any further questions you might have.

### Follow-Up Questions

1. How do you determine the vertex of a parabola given the focus and directrix?
2. What is the significance of the directrix in a parabola?
3. How do you calculate the distance from the vertex to the focus?
4. Can you explain why the parabola opens to the right in this case?
5. What are the different forms of the parabola equations depending on their orientation?
6. How does the value of \(p\) affect the shape of the parabola?
7. How can you find the length of the latus rectum of this parabola?
8. How would the equation change if the directrix were horizontal?

### Tip

When graphing parabolas, always identify the vertex, focus, and directrix first. This helps in understanding the orientation and shape of the parabola.

Learn how to graph a parabola with a given focus at (2, 2) and directrix x = -8. Understand how to find the vertex, orientation, and equation of the parabola. Suitable for high school students studying geometry and conic sections.

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