Find the equation of the parabola: . V(4,-4); F(4,-2) 3

To find the equation of a parabola given its vertex and focus, we can use the standard form of the equation for a parabola. Since the vertex \( V(4, -4) \) and focus \( F(4, -2) \) share the same x-coordinate, this indicates that the parabola opens either upwards or downwards.

### Step 1: Determine the direction of the parabola
Given the vertex and the focus:

- The vertex \( V(4, -4) \) is at \( (4, -4) \).
- The focus \( F(4, -2) \) is at \( (4, -2) \).

Since the focus is above the vertex (because \(-2 > -4\)), the parabola opens upwards.

### Step 2: Use the standard form of the parabola's equation
For a parabola that opens upwards or downwards with its vertex at \( (h, k) \), the equation is:

\[
(y - k) = \frac{1}{4p}(x - h)^2
\]

Where:
- \( (h, k) \) is the vertex,
- \( p \) is the distance from the vertex to the focus.

### Step 3: Find the value of \( p \)
The distance between the vertex and the focus is \( p \). In this case:

\[
p = | -2 - (-4) | = | -2 + 4 | = 2
\]

### Step 4: Write the equation
Substitute the vertex \( (h, k) = (4, -4) \) and \( p = 2 \) into the standard form:

\[
(y + 4) = \frac{1}{4(2)}(x - 4)^2
\]

Simplify the equation:

\[
(y + 4) = \frac{1}{8}(x - 4)^2
\]

### Final Equation
The equation of the parabola is:

\[
y + 4 = \frac{1}{8}(x - 4)^2
\]

Or equivalently:

\[
y = \frac{1}{8}(x - 4)^2 - 4
\]

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**Related Questions:**
1. How do you determine the direction in which a parabola opens based on its vertex and focus?
2. What is the significance of the parameter \( p \) in the equation of a parabola?
3. How can you find the equation of a parabola if you are given its directrix instead of the focus?
4. How does the distance from the vertex to the focus affect the "width" of the parabola?
5. What is the general process for deriving the equation of a parabola in different orientations (e.g., horizontal, vertical)?

**Tip:** Always ensure the orientation of the parabola by checking the relative position of the vertex and focus before writing the equation.

Learn how to find the equation of a parabola given its vertex and focus with a step-by-step solution. This problem involves using the standard form of the parabola equation, understanding the direction based on vertex and focus positions, and calculating the parameter 'p'. Suitable for high school students.

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