Make a table of values and sketch the graph of the equation. \[ y=x^{2}-1 \]

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 \( y = x^{2} - 1 \).