Determine the a) vertex,b)fucos,c)directrix,d)axis of symmetry and e)opening of the parabola in each item and draw the graph show the equetion
1.) x² = -4y
2.) 3y² = 24x
3.)(y + 5)² = -2(x - 9)

To solve the problems regarding parabolas, we need to identify the form of each equation, extract the required information (vertex, focus, directrix, axis of symmetry, and the direction of opening), and sketch the graph. Let's go step by step.

### Equation 1: \( x^2 = -4y \)
This equation is in the form \( x^2 = 4py \), which represents a vertical parabola. Here, \( 4p = -4 \), so \( p = -1 \).

#### a) Vertex:
- The vertex is at the origin, \( (0, 0) \), since there are no horizontal or vertical translations.

#### b) Focus:
- The focus is at \( (0, p) = (0, -1) \).

#### c) Directrix:
- The directrix is the line \( y = -p \), which is \( y = 1 \).

#### d) Axis of symmetry:
- The axis of symmetry is the vertical line \( x = 0 \) (the y-axis).

#### e) Opening:
- The parabola opens **downward** because \( p < 0 \).

### Graph:
The graph of the parabola would look like a U-shaped curve that opens downward with its vertex at the origin.

---

### Equation 2: \( 3y^2 = 24x \)
First, rewrite the equation in standard form:
\[
y^2 = \frac{24}{3}x = 8x
\]
This is in the form \( y^2 = 4px \), where \( 4p = 8 \), so \( p = 2 \).

#### a) Vertex:
- The vertex is at the origin, \( (0, 0) \), because there are no horizontal or vertical shifts.

#### b) Focus:
- The focus is at \( (p, 0) = (2, 0) \).

#### c) Directrix:
- The directrix is the vertical line \( x = -p \), which is \( x = -2 \).

#### d) Axis of symmetry:
- The axis of symmetry is the horizontal line \( y = 0 \) (the x-axis).

#### e) Opening:
- The parabola opens **to the right** because \( p > 0 \).

### Graph:
The graph would show a sideways parabola opening to the right, with its vertex at the origin.

---

### Equation 3: \( (y + 5)^2 = -2(x - 9) \)
This is a horizontal parabola equation in the form \( (y - k)^2 = 4p(x - h) \), where \( (h, k) \) is the vertex. Here, \( 4p = -2 \), so \( p = -\frac{1}{2} \), and the vertex is \( (h, k) = (9, -5) \).

#### a) Vertex:
- The vertex is \( (9, -5) \).

#### b) Focus:
- Since \( p = -\frac{1}{2} \), the focus is \( \left( 9 - \frac{1}{2}, -5 \right) = \left( \frac{17}{2}, -5 \right) \).

#### c) Directrix:
- The directrix is the line \( x = 9 + \frac{1}{2} = \frac{19}{2} \).

#### d) Axis of symmetry:
- The axis of symmetry is the horizontal line \( y = -5 \).

#### e) Opening:
- The parabola opens **to the left** because \( p < 0 \).

### Graph:
The graph would be a sideways parabola opening to the left, with the vertex at \( (9, -5) \).

---

### Summary:

1. **Equation 1**: \( x^2 = -4y \)
   - Vertex: \( (0, 0) \)
   - Focus: \( (0, -1) \)
   - Directrix: \( y = 1 \)
   - Axis of symmetry: \( x = 0 \)
   - Opens **downward**

2. **Equation 2**: \( 3y^2 = 24x \)
   - Vertex: \( (0, 0) \)
   - Focus: \( (2, 0) \)
   - Directrix: \( x = -2 \)
   - Axis of symmetry: \( y = 0 \)
   - Opens **to the right**

3. **Equation 3**: \( (y + 5)^2 = -2(x - 9) \)
   - Vertex: \( (9, -5) \)
   - Focus: \( \left( \frac{17}{2}, -5 \right) \)
   - Directrix: \( x = \frac{19}{2} \)
   - Axis of symmetry: \( y = -5 \)
   - Opens **to the left**

---

If you'd like, I can graph these parabolas for you as well.

### Would you like more details on any part of the explanation?

### Five related questions:
1. How do you find the equation of a parabola from its vertex and focus?
2. What are the properties of parabolas in real-world applications?
3. How can you determine if a parabola opens up, down, left, or right?
4. How do you calculate the focal length of a parabola from its equation?
5. What is the relationship between a parabola’s focus and directrix?

### Tip:
Always check the sign of \( p \) to determine the direction in which the parabola opens.

Determine the a) vertex, b) focus, c) directrix, d) axis of symmetry, and e) opening of the parabola in each item and draw the graph. 1.) x² = -4y 2.) 3y² = 24x 3.)(y + 5)² = -2(x - 9)

Learn how to find the vertex, focus, directrix, axis of symmetry, and the direction of opening for parabolas from equations such as x² = -4y, 3y² = 24x, and (y + 5)² = -2(x - 9). Includes step-by-step solutions and graphing examples. Suitable for high school students.

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