https://file.aiquickdraw.com/mathbot/file/1726412072711ts5kdj8v.jpg

find the length of latus rectum, focal length, opening of the graph, standard form, vertex, focus, endpoint of latus rectum, axis of symmetry, and directix  of the given parabola equation.

The equation from the image (Equation 3) is:

\[
y^2 + 16x - 2y + 49 = 0
\]

To analyze the properties of this parabola (focal length, latus rectum, focus, etc.), let's first rewrite it in standard form by completing the square.

### Step 1: Group the terms with \( y \) on one side

We start with:

\[
y^2 - 2y + 16x + 49 = 0
\]

Move the \( x \) and constant terms to the other side:

\[
y^2 - 2y = -16x - 49
\]

### Step 2: Complete the square on \( y \)

To complete the square, take the coefficient of \( y \), divide by 2, and square it. The coefficient of \( y \) is \( -2 \), so:

\[
\left(\frac{-2}{2}\right)^2 = 1
\]

Add 1 to both sides:

\[
y^2 - 2y + 1 = -16x - 49 + 1
\]

\[
(y - 1)^2 = -16x - 48
\]

Factor the right-hand side:

\[
(y - 1)^2 = -16(x + 3)
\]

### Step 3: Identify the standard form of the parabola

The equation is now in the standard form of a horizontal parabola:

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

Comparing \( (y - 1)^2 = -16(x + 3) \) with \( (y - k)^2 = 4p(x - h) \), we get:

- \( h = -3 \)
- \( k = 1 \)
- \( 4p = -16 \Rightarrow p = -4 \)

### Step 4: Analyze the properties of the parabola

- **Vertex**: The vertex is \( (h, k) = (-3, 1) \).
- **Focus**: The focus is at \( (h + p, k) = (-3 + (-4), 1) = (-7, 1) \).
- **Directrix**: The directrix is the vertical line \( x = h - p = -3 + 4 = 1 \).
- **Axis of symmetry**: Since the parabola opens horizontally, the axis of symmetry is \( y = 1 \).
- **Focal length**: The focal length is \( |p| = 4 \).
- **Length of latus rectum**: The length of the latus rectum is \( 4|p| = 16 \).
- **Endpoints of the latus rectum**: These points are \( (-7, 1 \pm 8) \), which gives the endpoints as \( (-7, 9) \) and \( (-7, -7) \).

### Summary

- **Standard form**: \( (y - 1)^2 = -16(x + 3) \)
- **Vertex**: \( (-3, 1) \)
- **Focus**: \( (-7, 1) \)
- **Directrix**: \( x = 1 \)
- **Axis of symmetry**: \( y = 1 \)
- **Focal length**: 4
- **Length of latus rectum**: 16
- **Endpoints of latus rectum**: \( (-7, 9) \) and \( (-7, -7) \)

Would you like more details or have any questions about these steps?

Here are 5 related questions:
1. How do you find the focus of a vertical parabola?
2. What does the sign of \( p \) indicate about the direction a parabola opens?
3. How do you complete the square for an equation with both \( x \) and \( y \) terms?
4. What is the significance of the latus rectum in a parabola?
5. How does the standard form of a parabola help in graphing?

**Tip**: Always check whether a parabola opens horizontally or vertically by examining which variable is squared.

Find the length of latus rectum, focal length, opening of the graph, standard form, vertex, focus, endpoint of latus rectum, axis of symmetry, and directrix of the given parabola equation.

Learn how to find the vertex, focus, directrix, axis of symmetry, and other properties of the parabola y^2 + 16x - 2y + 49 = 0. This step-by-step guide covers completing the square and analyzing conic sections. Suitable for students in Grades 9-12.

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