I. Graphing Parabolas: Graph the following equations in a rectangular coordinate plane. Then, identify the required properties being asked in each item.

1. 2x^2 - 8x - 16y - 24 = 0
Standard Form:
Opening:
Vertex: 
Distance between the vertex and focus:
Focus:
Equation of Axis of Symmetry: 
Length of Lactus Rectum:
Endpoints of Latus Rectum:
Equation of Directrix:

Let's start by transforming the given equation \( 2x^2 - 8x - 16y - 24 = 0 \) into a more recognizable form of a parabola equation, such as the vertex form. Then, we will extract the properties you need.

### Step 1: Rewriting the equation
The given equation is \( 2x^2 - 8x - 16y - 24 = 0 \).

We first isolate the \( y \)-terms:
\[
2x^2 - 8x = 16y + 24
\]
\[
2(x^2 - 4x) = 16y + 24
\]
Now, divide both sides by 2:
\[
x^2 - 4x = 8y + 12
\]

### Step 2: Completing the square
To rewrite the \( x^2 - 4x \) as a perfect square, complete the square:
\[
x^2 - 4x = (x - 2)^2 - 4
\]
So, the equation becomes:
\[
(x - 2)^2 - 4 = 8y + 12
\]
Add 4 to both sides:
\[
(x - 2)^2 = 8y + 16
\]
Finally, rewrite this as:
\[
(x - 2)^2 = 8(y + 2)
\]

This is now in the form of a parabola \( (x - h)^2 = 4p(y - k) \), where:
- \( h = 2 \)
- \( k = -2 \)
- \( 4p = 8 \), so \( p = 2 \).

### Step 3: Identifying properties

1. **Standard Form:**  
   \[
   (x - 2)^2 = 8(y + 2)
   \]
   This is a parabola that opens **vertically** because the squared term is \( (x - 2)^2 \), and the coefficient of \( y \) is positive.

2. **Opening:**  
   Since \( p = 2 > 0 \), the parabola opens **upward**.

3. **Vertex:**  
   The vertex of the parabola is at \( (h, k) = (2, -2) \).

4. **Distance between the vertex and focus:**  
   The distance between the vertex and the focus is \( p = 2 \).

5. **Focus:**  
   The focus lies **above** the vertex (since the parabola opens upward) at a distance \( p = 2 \). So, the focus is at:
   \[
   (2, -2 + 2) = (2, 0)
   \]

6. **Equation of Axis of Symmetry:**  
   The axis of symmetry is the vertical line passing through the vertex, which is:
   \[
   x = 2
   \]

7. **Length of Latus Rectum:**  
   The length of the latus rectum is \( 4p = 8 \).

8. **Endpoints of Latus Rectum:**  
   The latus rectum is horizontal, centered at the focus. The endpoints are found by moving \( \frac{8}{2} = 4 \) units to the left and right of the focus:
   \[
   (2 - 4, 0) = (-2, 0) \quad \text{and} \quad (2 + 4, 0) = (6, 0)
   \]
   So, the endpoints of the latus rectum are \( (-2, 0) \) and \( (6, 0) \).

9. **Equation of Directrix:**  
   The directrix is a horizontal line below the vertex at a distance \( p = 2 \). So, the equation of the directrix is:
   \[
   y = -2 - 2 = -4
   \]

### Summary:
- **Standard Form:** \( (x - 2)^2 = 8(y + 2) \)
- **Opening:** Upward
- **Vertex:** \( (2, -2) \)
- **Distance between the vertex and focus:** 2
- **Focus:** \( (2, 0) \)
- **Equation of Axis of Symmetry:** \(

Graph the following equations in a rectangular coordinate plane. Then, identify the required properties being asked in each item.
1. 2x^2 - 8x - 16y - 24 = 0
Standard Form:
Opening:
Vertex:
Distance between the vertex and focus:
Focus:
Equation of Axis of Symmetry:
Length of Latus Rectum:
Endpoints of Latus Rectum:
Equation of Directrix:

Learn how to graph the parabola equation 2x^2 - 8x - 16y - 24 in a rectangular coordinate plane. This guide covers key concepts such as identifying the vertex, focus, axis of symmetry, and directrix, with detailed step-by-step solutions. Suitable for students in Grades 9-12.

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