Graph each parabola. Include the vertex and at least three precise points on each side of the line of symmetry. Then describe the transformation in words. 27. $$ f(x)=x^{2} $$

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To graph the parabola $$ f(x) = x^{2} $$, follow these steps:

### 1. Identify the Vertex
- **Vertex**: The vertex form of a parabola is $$ f(x) = a(x-h)^{2} + k $$, where $$(h, k)$$ is the vertex.
- For $$ f(x) = x^{2} $$, comparing to the vertex form, we have $$ h = 0 $$ and $$ k = 0 $$.
- **Vertex**: $$(0, 0)$$

### 2. Determine the Line of Symmetry
- The line of symmetry passes through the vertex.
- **Line of Symmetry**: $$ x = 0 $$

### 3. Plot Points on Each Side of the Line of Symmetry
Choose values of $$ x $$ on both sides of the line of symmetry and calculate the corresponding $$ y $$-values.

#### Points to the **Left** of the Line of Symmetry ($$ x < 0 $$):
1. **$$ x = -3 $$**:
   $$ f(-3) = (-3)^{2} = 9 $$
   - Point: $$(-3, 9)$$

2. **$$ x = -2 $$**:
   $$ f(-2) = (-2)^{2} = 4 $$
   - Point: $$(-2, 4)$$

3. **$$ x = -1 $$**:
   $$ f(-1) = (-1)^{2} = 1 $$
   - Point: $$(-1, 1)$$

#### Points to the **Right** of the Line of Symmetry ($$ x > 0 $$):
1. **$$ x = 1 $$**:
   $$ f(1) = (1)^{2} = 1 $$
   - Point: $$(1, 1)$$

2. **$$ x = 2 $$**:
   $$ f(2) = (2)^{2} = 4 $$
   - Point: $$(2, 4)$$

3. **$$ x = 3 $$**:
   $$ f(3) = (3)^{2} = 9 $$
   - Point: $$(3, 9)$$

### 4. Sketch the Graph
Plot the vertex and the six calculated points on a coordinate plane. Draw a smooth, symmetric U-shaped curve through these points to form the parabola.

```
     |
   9 |        *           *
     |
   4 |     *     *     *     *
     |
   1 |  *           *           *
     |
-----+---------------------------
     |          0
```

*(Note: The above is a simplified representation. For an accurate graph, use graphing paper or graphing software.)*

### 5. Describe the Transformation
- **Transformation**: The equation $$ f(x) = x^{2} $$ represents the **parent quadratic function**.
- **Description**: This is the standard parabola with its vertex at the origin $$(0, 0)$$ and no transformations applied. It opens **upward** and is symmetric about the y-axis.

### Summary
- **Vertex**: $$(0, 0)$$
- **Line of Symmetry**: $$ x = 0 $$
- **Points**:
  - Left Side: $$(-3, 9)$$, $$(-2, 4)$$, $$(-1, 1)$$
  - Right Side: $$(1, 1)$$, $$(2, 4)$$, $$(3, 9)$$
- **Transformation**: No transformation; the standard upward-opening parabola centered at the origin.
To graph the parabola $$ f(x) = x^{2} $$: 1. **Vertex**: $$(0, 0)$$ 2. **Line of Symmetry**: $$ x = 0 $$ 3. **Points**: - Left Side: $$(-3, 9)$$, $$(-2, 4)$$, $$(-1, 1)$$ - Right Side: $$(1, 1)$$, $$(2, 4)$$, $$(3, 9)$$ 4. **Transformation**: No transformation; it's the standard upward-opening parabola centered at the origin. Graph the vertex and these points, then draw a symmetric U-shaped curve through them.

Graph each parabola. Include the vertex and at least three precise points on each side of the line of symmetry. Then describe the transformation in words. 27. \( f(x)=x^{2} \)

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