(x+4)²=12(y-3) find the coordinates show solution

To solve the equation \((x + 4)^2 = 12(y - 3)\) and find the coordinates, we first need to recognize that this is the equation of a parabola in the standard form \((x - h)^2 = 4p(y - k)\). 

### Step-by-Step Solution:

1. **Rewrite the equation**:
   \[(x + 4)^2 = 12(y - 3)\]

   This can be rewritten as:
   \[(x - (-4))^2 = 12(y - 3)\]

   Comparing this with the standard form \((x - h)^2 = 4p(y - k)\), we identify:
   - \(h = -4\)
   - \(k = 3\)
   - \(4p = 12\), so \(p = \frac{12}{4} = 3\)

2. **Interpret the parameters**:
   - The vertex of the parabola is at \((h, k) = (-4, 3)\).
   - The value of \(p = 3\) indicates that the distance from the vertex to the focus is 3 units, and since the parabola opens upwards (because \(p > 0\)), the focus will be 3 units above the vertex.

3. **Find the focus**:
   - The focus of the parabola is located at \((h, k + p)\).
   - Substituting the values we found:
     \[
     \text{Focus} = (-4, 3 + 3) = (-4, 6)
     \]

4. **Find the directrix**:
   - The directrix is a horizontal line given by \(y = k - p\).
   - Substituting the values:
     \[
     \text{Directrix} = y = 3 - 3 = 0
     \]

### Summary of coordinates:
- **Vertex**: \((-4, 3)\)
- **Focus**: \((-4, 6)\)
- **Directrix**: \(y = 0\)

Would you like more details or have any questions?

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Here are 8 related questions to consider:
1. How do you determine the direction in which a parabola opens?
2. What does the value of \(p\) signify in the context of a parabola?
3. How can you find the length of the latus rectum of the parabola?
4. How do you convert the standard form of a parabola to the vertex form?
5. What are the coordinates of the vertex if the equation is \(y = (x - 2)^2 + 5\)?
6. How does changing the value of \(p\) affect the shape of the parabola?
7. What is the axis of symmetry for the parabola given in the problem?
8. How would the equation change if the parabola opened to the left instead?

**Tip**: When dealing with parabolas, always identify the vertex first; it's the key to understanding the parabola's orientation and shape.

Learn how to find the coordinates (vertex, focus, and directrix) of the parabola given by the equation (x + 4)^2 = 12(y - 3). This problem involves interpreting the standard form of a parabola and identifying key points such as the vertex, focus, and directrix. Suitable for high school students.

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