(x-3)^2= -12(y+1)
type of parabola
opening of parabola
vertex
focus
directrix
axis of symmetry

Given the equation \((x - 3)^2 = -12(y + 1)\), we can determine the following characteristics of the parabola:

### 1. **Type of Parabola:**
The equation is in the form \((x - h)^2 = 4p(y - k)\), which represents a **parabola that opens vertically** (either up or down). Since the coefficient of \((y + 1)\) is negative, this parabola opens **downwards**.

### 2. **Opening of Parabola:**
The parabola opens **downward** because of the negative coefficient in front of the \((y + 1)\) term.

### 3. **Vertex:**
The vertex of the parabola is given by \((h, k)\). From the equation \((x - 3)^2 = -12(y + 1)\), we have:
- \(h = 3\)
- \(k = -1\)

So, the vertex is \((3, -1)\).

### 4. **Focus:**
The focus is found using the value of \(p\), which is determined from the equation \(4p = -12\). Solving for \(p\):
\[
p = \frac{-12}{4} = -3
\]
Since the parabola opens downward, the focus will be \(p\) units below the vertex. Therefore, the focus is at \((3, -1 - 3) = (3, -4)\).

### 5. **Directrix:**
The directrix is a horizontal line \(p\) units above the vertex, so:
\[
y = -1 + 3 = 2
\]
The directrix is the line \(y = 2\).

### 6. **Axis of Symmetry:**
The axis of symmetry is the vertical line that passes through the vertex, given by \(x = h\). Therefore, the axis of symmetry is:
\[
x = 3
\]

---

Would you like a detailed explanation of any specific part? Here are some related questions for further exploration:

1. How do you find the length of the latus rectum for a parabola?
2. What is the significance of the focus and directrix in a parabola?
3. How would the graph change if the coefficient of \((y + 1)\) were positive?
4. Can you derive the general formula for a vertically opening parabola?
5. How would you transform this equation into its standard form?

**Tip:** For any conic section, understanding the vertex form of its equation makes identifying the key properties, like focus and directrix, much easier.

Explore how to determine the type, opening, vertex, focus, directrix, and axis of symmetry of a parabola using an example equation (x - 3)^2 = -12(y + 1). This detailed explanation covers key concepts in parabola analysis. Ideal for high school students studying conic sections.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation