Let's solve the problem step by step: ### 1. **Understand the Function** The given function is: \[ g(x) = x^2 - 2 \] - This is a **quadratic function** (parabola). - The basic form is \( x^2 \), which is a parabola opening upwards with its vertex at the origin \((0,0)\). - The "-2" shifts the graph **downward by 2 units**. ### 2. **Domain** - The domain of \( g(x) = x^2 - 2 \) is **all real numbers** (\( -\infty < x < \infty \)). ### 3. **Vertex** - Vertex of \( x^2 \) is at (0,0). - For \( x^2 - 2 \), the vertex is at \( (0, -2) \). ### 4. **Axis of Symmetry** - The axis of symmetry is \( x = 0 \) (the y-axis). ### 5. **Table of Values** Let’s find a few points for plotting: | \( x \) | \( g(x) \) | |-------|----------------------| | -2 | \( (-2)^2 - 2 = 4 - 2 = 2 \) | | -1 | \( (-1)^2 - 2 = 1 - 2 = -1 \) | | 0 | \( 0^2 - 2 = -2 \) | | 1 | \( 1^2 - 2 = 1 - 2 = -1 \) | | 2 | \( 2^2 - 2 = 4 - 2 = 2 \) | ### 6. **Plotting the Points** Plot these points on the graph: - (-2, 2) - (-1, -1) - (0, -2) **[Vertex]** - (1, -1) - (2, 2) ### 7. **Sketch the Parabola** - Draw a smooth curve through these points, making sure the parabola opens upwards and is symmetric about the y-axis. --- ### **Final Answer (Graph Description)** - A parabola opening **upwards**. - **Vertex at (0, -2)** - **Axis of symmetry:** \(x = 0\) - **Domain:** All real numbers (\(-\infty, \infty\)) - Passes through points: \((-2, 2), (-1, -1), (0, -2), (1, -1), (2, 2)\) --- Here's a quick sketch of the graph: ``` y ↑ 10| 9| 8| 7| 6| 5| 4| 3| 2|* * 1| 0| -1| * * -2| * -3| -4| -5|____________________________→ x -2 -1 0 1 2 ``` Let me know if you need a graph drawn digitally or have further questions!