Graph the function. \[ f(x)=-3 x^{2} \]

To graph the function \( f(x) = -3x^{2} \), let's analyze its key features and plot some important points. ### 1. **Basic Shape** - **Parabola**: The function is a quadratic function, which graphs as a parabola. - **Orientation**: Since the coefficient of \( x^2 \) is negative (\(-3\)), the parabola opens **downward**. - **Vertex**: The vertex form of a parabola \( f(x) = a(x-h)^2 + k \) shows that the vertex is at \((h, k)\). For \( f(x) = -3x^{2} \), the vertex is at \((0, 0)\). ### 2. **Axis of Symmetry** - The axis of symmetry is the vertical line that passes through the vertex. For this function, it's the **y-axis** or \( x = 0 \). ### 3. **Plotting Key Points** Let's calculate the function's value at several \( x \)-values to plot points: | \( x \) | \( f(x) = -3x^{2} \) | |---------|-----------------------| | -2 | \( -3(-2)^2 = -12 \) | | -1 | \( -3(-1)^2 = -3 \) | | 0 | \( -3(0)^2 = 0 \) | | 1 | \( -3(1)^2 = -3 \) | | 2 | \( -3(2)^2 = -12 \) | ### 4. **Graphing the Function** Here's a simple **ASCII representation** of the graph: ``` y | 12| | 6| | --------|-------- x | -6| * * | -12|* * | ``` **Explanation of the ASCII Graph:** - The vertex \((0, 0)\) is at the center. - Points \((1, -3)\) and \((-1, -3)\) are marked with `*`. - Points \((2, -12)\) and \((-2, -12)\) are also marked, showing the parabola getting narrower as it moves away from the vertex. ### 5. **Graph Characteristics** - **Vertex**: \((0, 0)\) – the highest point of the parabola since it opens downward. - **Direction**: Opens downward. - **Width**: Steeper than the standard parabola \( y = -x^{2} \) due to the coefficient \(-3\). ### 6. **Visualization Using Desmos** For a more precise and visual graph, you can use graphing tools like [Desmos](https://www.desmos.com/calculator). Simply enter the function \( f(x) = -3x^{2} \) into the calculator to see an accurate graph.  *(Please note: In an actual implementation, ensure to include the graph image or provide a link to it. Since I cannot generate images directly, using Desmos or another graphing tool is recommended for visualization.)* ### Summary - **Type**: Downward-opening parabola - **Vertex**: (0, 0) - **Axis of Symmetry**: \( x = 0 \) - **Key Points**: \((-2, -12)\), \((-1, -3)\), \((0, 0)\), \((1, -3)\), \((2, -12)\) By plotting these points and understanding the shape, you can accurately graph the function \( f(x) = -3x^{2} \).