Identify the coordinates of the vertex and focus (y-2)= 8(x+3)., write the directrix and graph the parabola 

The given equation is \((y - 2) = 8(x + 3)\).

### Step 1: Rewrite the equation in standard form of a parabola

This equation represents a **horizontal parabola** of the form:
\[
(y - k)^2 = 4p(x - h)
\]
where \((h, k)\) is the vertex and \(p\) is the distance between the vertex and the focus.

Let's compare this equation to the standard form:
\[
(y - 2) = 8(x + 3)
\]
Rearrange to isolate \(x\):
\[
(y - 2)^2 = 8(x + 3)
\]
Now, we can see that:
- \(4p = 8\), which means \(p = 2\).
- The vertex \((h, k)\) is \((-3, 2)\).

### Step 2: Identify the vertex and focus
- The **vertex** is \((-3, 2)\).
- Since the parabola opens horizontally and \(p = 2\), the focus is located \(p\) units to the right of the vertex (for a parabola that opens to the right, \(p\) is positive).

Thus, the focus is \((-3 + 2, 2)\) = \((-1, 2)\).

### Step 3: Write the directrix
The directrix is a vertical line, located \(p = 2\) units to the left of the vertex. Therefore, the equation of the directrix is:
\[
x = -3 - 2 = -5
\]

### Summary:
- **Vertex**: \((-3, 2)\)
- **Focus**: \((-1, 2)\)
- **Directrix**: \(x = -5\)

### Step 4: Graph the parabola

To graph the parabola, follow these steps:
1. Plot the vertex \((-3, 2)\).
2. Plot the focus \((-1, 2)\).
3. Draw the directrix, which is a vertical line at \(x = -5\).
4. Sketch the parabola opening to the right (since it is a horizontal parabola with a positive value for \(p\)).

Would you like more details or a visual graph for better understanding?

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Here are five related questions to expand on this problem:
1. How does the value of \(p\) influence the shape of a parabola?
2. What would happen if the equation had been \((y - 2)^2 = -8(x + 3)\)?
3. How do you determine whether a parabola opens horizontally or vertically?
4. Can you explain the relationship between the vertex, focus, and directrix in parabolas?
5. How would the graph change if the focus were moved further from the vertex?

**Tip**: The directrix and focus are always equidistant from the vertex, forming the basis of the parabola’s symmetry.

Identify the coordinates of the vertex and focus (y-2)= 8(x+3). Write the directrix and graph the parabola.

Explore how to find the vertex, focus, and directrix of the parabola (y-2) = 8(x+3). This detailed solution covers key concepts in conic sections, including the equation of a parabola, focus, and directrix, with step-by-step instructions. Ideal for high school students (Grades 10-12).

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