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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UpStudy AI · Accepted Answer

Sure! Let’s graph the parabola $$ f(x) = x^{2} $$ step by step.

### 1. **Identify the Vertex**
The vertex of a parabola in the form $$ f(x) = a(x - h)^2 + k $$ is at the point $$ (h, k) $$.

For $$ f(x) = x^{2} $$:
- $$ h = 0 $$
- $$ k = 0 $$

**Vertex:** $$ (0, 0) $$

### 2. **Determine the Line of Symmetry**
The line of symmetry for a parabola $$ f(x) = a(x - h)^2 + k $$ is the vertical line $$ x = h $$.

For $$ f(x) = x^{2} $$:
- **Line of Symmetry:** $$ x = 0 $$ (the y-axis)

### 3. **Plot Additional Points**
We need at least three precise points on each side of the line of symmetry. Let’s choose $$ x $$-values symmetric around the vertex.

#### **Left Side of the Line of Symmetry**

| $$ x $$ | $$ f(x) = x^{2} $$ |
|---|---|
| -3 | $$ (-3)^2 = 9 $$ → $$ (-3, 9) $$ |
| -2 | $$ (-2)^2 = 4 $$ → $$ (-2, 4) $$ |
| -1 | $$ (-1)^2 = 1 $$ → $$ (-1, 1) $$ |

#### **Right Side of the Line of Symmetry**

| $$ x $$ | $$ f(x) = x^{2} $$ |
|---|---|
| 1 | $$ (1)^2 = 1 $$ → $$ (1, 1) $$ |
| 2 | $$ (2)^2 = 4 $$ → $$ (2, 4) $$ |
| 3 | $$ (3)^2 = 9 $$ → $$ (3, 9) $$ |

### 4. **Plotting the Points**
Plot the following points on a coordinate plane:

- **Vertex:** $$ (0, 0) $$
- **Left Side:** $$ (-3, 9) $$, $$ (-2, 4) $$, $$ (-1, 1) $$
- **Right Side:** $$ (1, 1) $$, $$ (2, 4) $$, $$ (3, 9) $$

### 5. **Drawing the Parabola**
Connect the points with a smooth, U-shaped curve opening upwards. Ensure the graph is symmetric about the y-axis (line of symmetry $$ x = 0 $$).

### 6. **Transformation Description**

For $$ f(x) = x^{2} $$:

- **Base Form:** This is the standard quadratic function.
- **Vertex:** Located at the origin $$ (0, 0) $$, indicating no vertical or horizontal shifts.
- **Line of Symmetry:** The y-axis ($$ x = 0 $$), showing the graph is symmetric around this line.
- **Opening Direction:** Opens upward since the coefficient of $$ x^{2} $$ is positive.
- **Width:** The parabola has a standard width; since there’s no coefficient stretching or compressing it, it matches the basic $$ x^{2} $$ shape.

**In Summary:** The graph of $$ f(x) = x^{2} $$ is the standard parabola with its vertex at the origin, symmetric about the y-axis, opening upwards, and passing through the points $$ (-3, 9) $$, $$ (-2, 4) $$, $$ (-1, 1) $$, $$ (1, 1) $$, $$ (2, 4) $$, and $$ (3, 9) $$.

### **Graphical Representation**

While I can't provide a visual graph here, you can plot the points mentioned above on graph paper or using graphing software to visualize the parabola.

**Example:**

```
y
|
9 |        *               *
8 |        
7 |
6 |
5 |
4 |      *          *
3 |
2 |
1 |    *      *      *
0 |-------------------------- x
   -3  -2  -1   0   1   2   3
```

- The asterisks (*) represent the plotted points.
- Connect them smoothly to form the U-shaped parabola.
To graph the parabola $$ f(x) = x^{2} $$: 1. **Vertex:** $$ (0, 0) $$ 2. **Line of Symmetry:** $$ x = 0 $$ (y-axis) 3. **Points:** - Left Side: $$ (-3, 9) $$, $$ (-2, 4) $$, $$ (-1, 1) $$ - Right Side: $$ (1, 1) $$, $$ (2, 4) $$, $$ (3, 9) $$ 4. **Transformation:** The graph is symmetric about the y-axis, opens upwards, and has its vertex at the origin. **Graph Description:** The parabola $$ f(x) = x^{2} $$ is symmetric around the y-axis, with its lowest point at the origin. It opens upwards and passes through the points $$ (-3, 9) $$, $$ (-2, 4) $$, $$ (-1, 1) $$, $$ (1, 1) $$, $$ (2, 4) $$, and $$ (3, 9) $$.

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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