Graph the parabola (pre calculus)  1. Vertex: (3,2) Focus: (3,0) 2. Vertex: (0,2) Focus: (0,5) give the Directrix, Length of LR, Endpoints and axis of symmetry

Here are the graphs for the two parabolas based on the given conditions:

### **Case 1**: Vertex (3, 2) and Focus (3, 0)
- **Equation**: \( (x - 3)^2 = -8(y - 2) \)
- **Directrix**: \( y = 4 \)
- **Length of Latus Rectum**: 8
- **Endpoints of Latus Rectum**: (7, 0) and (-1, 0)
- **Axis of Symmetry**: \( x = 3 \)

### **Case 2**: Vertex (0, 2) and Focus (0, 5)
- **Equation**: \( x^2 = 12(y - 2) \)
- **Directrix**: \( y = -1 \)
- **Length of Latus Rectum**: 12
- **Endpoints of Latus Rectum**: (6, 5) and (-6, 5)
- **Axis of Symmetry**: \( x = 0 \)

Let me know if you need further clarification or additional details!

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Here are 8 related questions you can explore:
1. What happens to the parabola if the directrix is moved closer to the vertex?
2. How does the focus determine the direction a parabola opens?
3. How do you find the vertex from the general equation of a parabola?
4. What is the significance of the latus rectum in determining the width of a parabola?
5. How do the equations of horizontal and vertical parabolas differ?
6. How do changes in \( p \) (distance to focus) affect the shape of a parabola?
7. How can you determine the equation of the axis of symmetry from the equation of the parabola?
8. What are some real-world applications of parabolas?

**Tip**: The distance between the focus and the vertex, \( p \), controls how "wide" or "narrow" the parabola is. A larger \( |p| \) results in a wider parabola.

Explore how to graph parabolas given vertex and focus points. Learn to find the directrix, length of the latus rectum, endpoints, and axis of symmetry for different scenarios. Detailed solutions and explanations provided.

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