Determine the vertex, focus, directrix, ends of latus rectum, and axis of symmetry of y^2 + 4x - 14y + 53 = 0

To determine the vertex, focus, directrix, ends of the latus rectum, and axis of symmetry for the given equation \( y^2 + 4x - 14y + 53 = 0 \), we'll follow these steps:

### 1. **Rewrite the Equation**:
The given equation is:
\[
y^2 + 4x - 14y + 53 = 0
\]
First, we'll rearrange the equation to group the \( y \)-terms together and complete the square.

\[
y^2 - 14y + 4x + 53 = 0
\]
\[
y^2 - 14y = -4x - 53
\]

### 2. **Complete the Square**:
To complete the square for the \( y \)-terms, take half of the coefficient of \( y \), square it, and add it to both sides:

\[
y^2 - 14y + \left(\frac{-14}{2}\right)^2 = -4x - 53 + \left(\frac{-14}{2}\right)^2
\]
\[
y^2 - 14y + 49 = -4x - 53 + 49
\]
\[
(y - 7)^2 = -4x - 4
\]
\[
(y - 7)^2 = -4(x + 1)
\]

### 3. **Identify the Standard Form**:
Now, the equation is in the standard form of a parabola that opens leftward or rightward:
\[
(y - k)^2 = 4p(x - h)
\]
Here, we have:
- \( h = -1 \)
- \( k = 7 \)
- \( 4p = -4 \)

### 4. **Determine \( p \)**:
To find \( p \):
\[
4p = -4 \quad \Rightarrow \quad p = \frac{-4}{4} = -1
\]

### 5. **Vertex**:
The vertex \((h, k)\) is:
\[
\text{Vertex: } (-1, 7)
\]

### 6. **Focus**:
The focus is located \( p \) units from the vertex in the direction the parabola opens. Since \( p = -1 \) and the parabola opens to the left:
\[
\text{Focus: } (h + p, k) = (-1 - 1, 7) = (-2, 7)
\]

### 7. **Directrix**:
The directrix is a vertical line located \( p \) units away from the vertex in the opposite direction of the focus. So, the directrix is:
\[
\text{Directrix: } x = h - p = -1 - (-1) = 0
\]

### 8. **Ends of Latus Rectum**:
The latus rectum is a line segment perpendicular to the axis of symmetry and passes through the focus. The length of the latus rectum is \( |4p| = 4 \).

Since the latus rectum is horizontal and centered at the focus, the ends of the latus rectum are:
\[
\text{Ends of latus rectum: } (-2, 7 + 2) = (-2, 9) \quad \text{and} \quad (-2, 7 - 2) = (-2, 5)
\]

### 9. **Axis of Symmetry**:
The axis of symmetry is a horizontal line passing through the vertex (and the focus). Hence:
\[
\text{Axis of Symmetry: } y = 7
\]

### **Summary**:
- **Vertex**: \((-1, 7)\)
- **Focus**: \((-2, 7)\)
- **Directrix**: \(x = 0\)
- **Ends of Latus Rectum**: \((-2, 9)\) and \((-2, 5)\)
- **Axis of Symmetry**: \(y = 7\)

Would you like any further explanation or have any other questions on this problem?

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Here are 5 questions that relate to this topic:
1. How would the equation change if the parabola opened to the right instead of the left?
2. How do the values of \( h \) and \( k \) influence the graph of the parabola?
3. What would the equation of the parabola be if it were translated 3 units down?
4. How can you determine the direction of opening for a parabola just by looking at the equation?
5. What is the significance of the directrix in relation to the focus of a parabola?

**Tip:** Completing the square is a crucial step in converting quadratic equations into standard form, allowing for easy identification of key features like the vertex, focus, and directrix.

Learn how to determine the vertex, focus, directrix, and ends of latus rectum for the parabola given by the equation y^2 + 4x - 14y + 53 = 0. This problem involves using vertex form of a parabola, completing the square, and identifying key geometric features. Suitable for advanced high school students and calculus learners.

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