4. Graph the parabola. Include the vertex and two accurate points on each side of the axis of symmetry. $$ f(x)=2(x-3)^{2}-6 $$

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To graph the parabola defined by the function $$ f(x) = 2(x - 3)^2 - 6 $$, we'll follow these steps:

1. **Identify the Vertex**
2. **Determine the Axis of Symmetry**
3. **Calculate Additional Points**
4. **Plot the Points and Draw the Parabola**

Let's go through each step in detail.

---

### 1. Identify the Vertex

The given quadratic function is in **vertex form**, which is:

$$ f(x) = a(x - h)^2 + k $$

Here, $$(h, k)$$ represents the vertex of the parabola.

Comparing with the given function:

$$ f(x) = 2(x - 3)^2 - 6 $$

- $$ h = 3 $$
- $$ k = -6 $$

**Vertex:** $$ (3, -6) $$

---

### 2. Determine the Axis of Symmetry

The **axis of symmetry** is a vertical line that passes through the vertex. Its equation is:

$$ x = h $$

For our function:

$$ x = 3 $$

This line divides the parabola into two mirror-image halves.

---

### 3. Calculate Additional Points

To accurately graph the parabola, we'll find two points on each side of the axis of symmetry.

#### Selecting x-values:

Choose x-values around the vertex $$ x = 3 $$:

- **Left of Vertex:** $$ x = 1 $$ and $$ x = 2 $$
- **Right of Vertex:** $$ x = 4 $$ and $$ x = 5 $$

#### Calculating f(x) for Each x-value:

1. **For $$ x = 1 $$:**
   $$
   f(1) = 2(1 - 3)^2 - 6 = 2(4) - 6 = 8 - 6 = 2
   $$
   **Point:** $$ (1, 2) $$

2. **For $$ x = 2 $$:**
   $$
   f(2) = 2(2 - 3)^2 - 6 = 2(1) - 6 = 2 - 6 = -4
   $$
   **Point:** $$ (2, -4) $$

3. **For $$ x = 4 $$:**
   $$
   f(4) = 2(4 - 3)^2 - 6 = 2(1) - 6 = 2 - 6 = -4
   $$
   **Point:** $$ (4, -4) $$

4. **For $$ x = 5 $$:**
   $$
   f(5) = 2(5 - 3)^2 - 6 = 2(4) - 6 = 8 - 6 = 2
   $$
   **Point:** $$ (5, 2) $$

**Additional Points:**

- Left Side:
  - $$ (1, 2) $$
  - $$ (2, -4) $$

- Right Side:
  - $$ (4, -4) $$
  - $$ (5, 2) $$

---

### 4. Plot the Points and Draw the Parabola

Here's a table of the key points to plot:

| $$ x $$ | $$ f(x) $$ | Point      |
|---------|------------|------------|
| 1       | 2          | (1, 2)     |
| 2       | -4         | (2, -4)    |
| 3       | -6         | (3, -6) **Vertex** |
| 4       | -4         | (4, -4)    |
| 5       | 2          | (5, 2)     |

**Steps to Graph:**

1. **Draw the Coordinate Plane:**
   - Horizontal axis (x-axis) and vertical axis (y-axis).

2. **Plot the Vertex:**
   - Mark the point $$ (3, -6) $$.

3. **Draw the Axis of Symmetry:**
   - Draw a dashed vertical line through $$ x = 3 $$.

4. **Plot the Additional Points:**
   - Plot $$ (1, 2) $$, $$ (2, -4) $$, $$ (4, -4) $$, and $$ (5, 2) $$.

5. **Draw the Parabola:**
   - Connect the points smoothly, ensuring the parabola opens **upward** since the coefficient $$ a = 2 $$ is positive.
   - The graph should be symmetric about the axis $$ x = 3 $$.

---

### Visual Representation

While I can't provide a visual graph here, following the steps above with graph paper or a graphing tool will yield the correct parabola. Here's a rough sketch of how the points align:

```
y
|
2 |      *           *
1 |
0 |
-2|
-4 |   *       *    
-6 |      V
-8|
   ------------------------- x
    1   2   3   4   5
```

- **$$ V $$** represents the Vertex at $$ (3, -6) $$.
- Stars ($$*$$) represent the additional points.

---

### Summary of Key Features

- **Vertex:** $$ (3, -6) $$
- **Axis of Symmetry:** $$ x = 3 $$
- **Additional Points:**
  - Left: $$ (1, 2) $$ and $$ (2, -4) $$
  - Right: $$ (4, -4) $$ and $$ (5, 2) $$
- **Direction:** Opens **upward** (since $$ a = 2 > 0 $$)

By plotting these points and connecting them with a smooth, U-shaped curve, you'll accurately graph the parabola defined by $$ f(x) = 2(x - 3)^2 - 6 $$.
To graph the parabola $$ f(x) = 2(x - 3)^2 - 6 $$, plot the vertex at $$ (3, -6) $$ and the axis of symmetry at $$ x = 3 $$. Then, plot additional points such as $$ (1, 2) $$, $$ (2, -4) $$, $$ (4, -4) $$, and $$ (5, 2) $$. Draw a smooth, upward-opening curve through these points to complete the graph.

4. Graph the parabola. Include the vertex and two accurate points on each side of the axis of symmetry. \( f(x)=2(x-3)^{2}-6 \)

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