https://r2.erweima.ai/mathbot/image/1726265529715ufbzoil7.jpg

Let's analyze and solve the parabolas shown in the image. Based on the graphs and the given data, we can determine the vertex form of the equation for each parabola.

### 1) First parabola:
- **Vertex (V):** \((0, 0)\)
- **Focus (F):** \((3, 0)\)
- This parabola opens to the right, and the distance from the vertex to the focus is \(p = 3\).

The standard form of a horizontal parabola with vertex \((h, k)\) is:

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

Since the vertex is \((0, 0)\) and \(p = 3\), the equation becomes:

\[
y^2 = 12x
\]

---

### 2) Second parabola:
- **Vertex (V):** \((0, 3)\)
- **Focus (F):** \((0, 1)\)
- This parabola opens downwards, and the distance from the vertex to the focus is \(p = -2\) (since the parabola opens downwards).

The standard form of a vertical parabola with vertex \((h, k)\) is:

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

Since the vertex is \((0, 3)\) and \(p = -2\), the equation becomes:

\[
x^2 = -8(y - 3)
\]

---

### 3) Third parabola:
- **Vertex (V):** \((-2, 1)\)
- **Directrix (d):** \(x = -5\)
- The parabola opens to the right, and the distance from the vertex to the directrix is \(p = 3\) (since the directrix is to the left of the vertex).

The standard form of a horizontal parabola with vertex \((h, k)\) is:

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

Since the vertex is \((-2, 1)\) and \(p = 3\), the equation becomes:

\[
(y - 1)^2 = 12(x + 2)
\]

---

### Summary of the equations:
1) \( y^2 = 12x \)
2) \( x^2 = -8(y - 3) \)
3) \( (y - 1)^2 = 12(x + 2) \)

Would you like more details on any of these steps?

Here are 5 follow-up questions related to this problem:
1) How do you calculate the distance between the vertex and focus or directrix?
2) How do you determine if a parabola opens up, down, left, or right?
3) What is the general formula for the vertex form of a parabola?
4) Can you derive the equation of a parabola given only the focus and directrix?
5) How does the value of \(p\) affect the width of a parabola?

**Tip:** Remember that the sign of \(p\) determines the direction in which the parabola opens. Positive \(p\) means up or right, while negative \(p\) means down or left.

Determine the equation of each parabola from its graph.

Learn how to determine the equation of parabolas by analyzing their vertex, focus, and directrix. This problem involves applying the standard forms for horizontal and vertical parabolas, suitable for high school students.

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