Parabola; vertex, directrix, focus, axis of symmetry, lotus rectum 1 and 2 of y=(x+4)²-3

To analyze the given parabola \( y = (x + 4)^2 - 3 \), we can identify various key properties like the vertex, directrix, focus, axis of symmetry, and latus rectum (sometimes referred to as the length of the focal chord). Here's a breakdown:

### 1. **Vertex**:
The vertex form of a parabola is given by \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex.
- For the equation \( y = (x + 4)^2 - 3 \), we can rewrite it as:
  \[
  y = (x - (-4))^2 - 3
  \]
  Here, \( h = -4 \) and \( k = -3 \).
- **Vertex**: \((-4, -3)\)

### 2. **Axis of Symmetry**:
The axis of symmetry is a vertical line that passes through the vertex.
- The equation for the axis of symmetry is \( x = h \).
- **Axis of Symmetry**: \( x = -4 \)

### 3. **Focus**:
The focus of a parabola is given by the formula:
\[
\text{Focus} = \left(h, k + \frac{1}{4a}\right)
\]
where \( a \) is the coefficient of \((x-h)^2\) in the vertex form of the equation. In this case, \( a = 1 \) (since the equation is \( y = 1(x + 4)^2 - 3 \)).
- Plugging in the values:
  \[
  \text{Focus} = \left(-4, -3 + \frac{1}{4(1)}\right) = \left(-4, -3 + \frac{1}{4}\right) = \left(-4, -\frac{11}{4}\right)
  \]
- **Focus**: \((-4, -\frac{11}{4})\) or \((-4, -2.75)\)

### 4. **Directrix**:
The directrix is a horizontal line given by the formula:
\[
\text{Directrix} = y = k - \frac{1}{4a}
\]
- Plugging in the values:
  \[
  \text{Directrix} = y = -3 - \frac{1}{4(1)} = y = -\frac{13}{4} = y = -3.25
  \]
- **Directrix**: \( y = -\frac{13}{4} \) or \( y = -3.25 \)

### 5. **Latus Rectum**:
The length of the latus rectum is given by:
\[
\text{Length of Latus Rectum} = \frac{1}{|a|}
\]
- Here, \( a = 1 \), so:
  \[
  \text{Length of Latus Rectum} = \frac{1}{1} = 1
  \]
- The endpoints of the latus rectum are calculated by moving horizontally from the focus by half the length of the latus rectum:
  \[
  \text{Endpoints} = \left( -4 \pm \frac{1}{2}, -\frac{11}{4} \right) = \left( -4.5, -2.75 \right) \text{ and } \left( -3.5, -2.75 \right)
  \]
- **Latus Rectum 1**: \((-4.5, -2.75)\)
- **Latus Rectum 2**: \((-3.5, -2.75)\)

### Summary:
- **Vertex**: \((-4, -3)\)
- **Axis of Symmetry**: \(x = -4\)
- **Focus**: \((-4, -2.75)\)
- **Directrix**: \(y = -3.25\)
- **Latus Rectum**:
  - **Latus Rectum 1**: \((-4.5, -2.75)\)
  - **Latus Rectum 2**: \((-3.5, -2.75)\)

Would you like any additional details or have any questions?

### Further Questions:
1. How do you derive the formula for the focus of a parabola?
2. Can you explain the significance of the directrix in a parabola?
3. What happens to the parabola if we change the coefficient \(a\) in the equation?
4. How do we calculate the distance between the vertex and focus?
5. What are the geometric properties of the parabola related to the latus rectum?

### Tip:
The vertex form of a parabola \( y = a(x-h)^2 + k \) is useful for quickly identifying the vertex and analyzing transformations such as shifts and stretches.

Explore the properties of the parabola y = (x+4)^2 - 3 including its vertex, focus, directrix, axis of symmetry, and latus rectum. Learn how to derive these properties step-by-step with detailed explanations and examples.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation