Answer
To graph \( y = x^{2} - 1 \), create a table of values, plot the points, and draw a U-shaped parabola opening upwards with its vertex at \((0, -1)\).
Solution
Certainly! Let's create a table of values for the equation \( y = x^{2} - 1 \) and then discuss how to sketch its graph.
### Table of Values
Choose several \( x \)-values and compute the corresponding \( y \)-values.
| \( x \) | \( y = x^{2} - 1 \) |
|---------|---------------------|
| -3 | \( (-3)^2 - 1 = 9 - 1 = 8 \) |
| -2 | \( (-2)^2 - 1 = 4 - 1 = 3 \) |
| -1 | \( (-1)^2 - 1 = 1 - 1 = 0 \) |
| 0 | \( (0)^2 - 1 = 0 - 1 = -1 \) |
| 1 | \( (1)^2 - 1 = 1 - 1 = 0 \) |
| 2 | \( (2)^2 - 1 = 4 - 1 = 3 \) |
| 3 | \( (3)^2 - 1 = 9 - 1 = 8 \) |
### Plotting the Points
Using the table above, plot the points \((-3, 8)\), \((-2, 3)\), \((-1, 0)\), \((0, -1)\), \((1, 0)\), \((2, 3)\), and \((3, 8)\) on a coordinate plane.
### Sketching the Graph
The equation \( y = x^{2} - 1 \) represents a **parabola** that opens **upwards** because the coefficient of \( x^{2} \) is positive. Here's how to sketch it:
1. **Vertex**: The lowest point of the parabola is the vertex. For \( y = x^{2} - 1 \), the vertex is at \( (0, -1) \).
2. **Symmetry**: The parabola is symmetric about the **y-axis** (vertical line \( x = 0 \)) because the equation contains \( x^{2} \).
3. **Plot the Points**: Use the table to plot the points accurately.
4. **Draw the Parabola**: Connect the points smoothly to form the U-shaped curve.
### Visual Representation
While I can't draw an actual graph here, I can provide an approximate ASCII representation:
```
y
|
8 | * *
7 |
6 |
5 |
4 |
3 | * *
2 |
1 |
0 | * *
-1|-------------------------- x
-3 -2 -1 0 1 2 3
```
- The "*" symbols represent the plotted points.
- The vertex is at \((0, -1)\), which is the lowest point on the graph.
- The parabola opens upwards, passing through all the plotted points.
### Key Features of the Graph
- **Vertex**: \( (0, -1) \)
- **Axis of Symmetry**: \( x = 0 \) (the y-axis)
- **Direction**: Opens upwards
- **Y-intercept**: \( (0, -1) \)
- **X-intercepts**: \( (-1, 0) \) and \( (1, 0) \)
By plotting these points and understanding the shape of the parabola, you can accurately sketch the graph of \( y = x^{2} - 1 \).
Reviewed and approved by the UpStudy tutoring team
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