To graph the equation \((y + 5)^2 = 20(x + 2)\), let's break down the key elements:

### 1. **Identify the type of conic section:**
This equation is in the form \((y - k)^2 = 4p(x - h)\), which represents a **parabola** that opens either to the right or left, depending on the sign of the coefficient of the \((x - h)\) term.

### 2. **Determine the vertex:**
The vertex \((h, k)\) is given by:
- \(h = -2\)
- \(k = -5\)

So, the vertex of the parabola is \((-2, -5)\).

### 3. **Find the direction of opening:**
Since the coefficient of \((x + 2)\) is positive, the parabola opens **to the right**.

### 4. **Calculate the value of \(p\):**
From the equation \(4p = 20\), we find:
\[
p = \frac{20}{4} = 5
\]
This means the focus is 5 units to the right of the vertex.

### 5. **Plotting the key points:**
- **Vertex:** \((-2, -5)\)
- **Focus:** Located 5 units to the right of the vertex, so at \((3, -5)\).
- **Directrix:** A vertical line 5 units to the left of the vertex, which is \(x = -7\).
- **Axis of Symmetry:** The horizontal line \(y = -5\).

### 6. **Sketch the parabola:**
- Start by plotting the vertex at \((-2, -5)\).
- Draw the focus at \((3, -5)\).
- Draw the directrix as a vertical line at \(x = -7\).
- The parabola will curve outward to the right, with the vertex being the closest point to the directrix, and the curve getting wider as it moves away from the vertex.

Would you like me to generate a visual representation of this graph? If yes, I can create an image for you.

Here are some related questions for further exploration:

1. How do you determine the length of the latus rectum for this parabola?
2. What are the key differences between parabolas that open vertically and those that open horizontally?
3. How would the graph change if the coefficient of \((x + 2)\) were negative?
4. What are the general properties of a parabola's focus and directrix?
5. How would you find the equation of a parabola given its vertex and focus?

**Tip:** When graphing parabolas, always identify the vertex, focus, and directrix first. These elements help guide the shape and direction of the curve.

Learn how to graph the parabola defined by the equation (y + 5)^2 = 20(x + 2). This problem involves identifying the vertex, focus, and directrix of the parabola, suitable for high school students. Includes detailed steps and key concepts such as vertex form of a parabola and the distance formula.

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