Graph the function. $$ f(x)=-3 x^{2} $$

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UpStudy AI · Accepted Answer

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.

![Graph of f(x) = -3x²](https://i.imgur.com/ZhJ6V4X.png)

*(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 $$ f(x) = -3x^{2} $$.
To graph the function $$ f(x) = -3x^{2} $$, plot the vertex at (0,0), and draw a downward-opening parabola passing through points like (-2, -12), (-1, -3), (1, -3), and (2, -12).

Graph the function.

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