Question
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. \( f(x)=x^{2} \)
Ask by Norton Beck. in the United States
Feb 03,2025
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Answer
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 \( (3, 9) \).
Solution
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.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
The graph of the function \( f(x) = x^2 \) is a standard parabola that opens upwards. Its vertex is located at the origin (0, 0). To find other points, you can evaluate the function at \( x = -3, -2, -1, 1, 2, 3 \). This gives you the points: (-3, 9), (-2, 4), (-1, 1), (1, 1), (2, 4), and (3, 9). Each of these points helps demonstrate the symmetry around the vertical line of symmetry, \( x = 0 \). In terms of transformation, \( f(x) = x^2 \) represents a basic quadratic function that has not undergone any shifts, stretches, or compressions. It is the simplest form of a parabola, showcasing a perfect U-shape that opens upwards symmetrically around the y-axis.
