Graph the following function. $$ f(x)=-\frac{1}{4} x^{2} $$

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To graph the function $$ f(x) = -\frac{1}{4} x^{2} $$, we can follow these steps:

1. **Identify the type of function**: This is a quadratic function in the form $$ f(x) = ax^2 + bx + c $$, where $$ a = -\frac{1}{4} $$, $$ b = 0 $$, and $$ c = 0 $$. Since $$ a < 0 $$, the parabola opens downwards.

2. **Find the vertex**: The vertex of a quadratic function in standard form $$ ax^2 + bx + c $$ is given by the formula $$ x = -\frac{b}{2a} $$. Here, since $$ b = 0 $$:
   $$
   x = -\frac{0}{2 \cdot -\frac{1}{4}} = 0
   $$
   To find the y-coordinate of the vertex, substitute $$ x = 0 $$ into the function:
   $$
   f(0) = -\frac{1}{4}(0)^2 = 0
   $$
   So, the vertex is at the point $$ (0, 0) $$.

3. **Determine the axis of symmetry**: The axis of symmetry is the vertical line that passes through the vertex. For this function, it is $$ x = 0 $$.

4. **Find additional points**: To get a better shape of the parabola, calculate $$ f(x) $$ for a few values of $$ x $$:
   - For $$ x = 2 $$:
     $$
     f(2) = -\frac{1}{4}(2)^2 = -\frac{1}{4} \cdot 4 = -1
     $$
   - For $$ x = -2 $$:
     $$
     f(-2) = -\frac{1}{4}(-2)^2 = -\frac{1}{4} \cdot 4 = -1
     $$
   - For $$ x = 4 $$:
     $$
     f(4) = -\frac{1}{4}(4)^2 = -\frac{1}{4} \cdot 16 = -4
     $$
   - For $$ x = -4 $$:
     $$
     f(-4) = -\frac{1}{4}(-4)^2 = -\frac{1}{4} \cdot 16 = -4
     $$

5. **Plot the points**: The points we have are:
   - Vertex: $$ (0, 0) $$
   - Points: $$ (2, -1) $$, $$ (-2, -1) $$, $$ (4, -4) $$, $$ (-4, -4) $$

6. **Draw the graph**: Plot these points on a coordinate plane and draw a smooth curve through them, making sure the parabola opens downwards.

The graph of the function $$ f(x) = -\frac{1}{4} x^{2} $$ will look like a downward-opening parabola with its vertex at the origin (0, 0). The parabola will be symmetric about the y-axis.

If you were to visualize it, it would look something like this:

```
   |
  0|         *
   |       *   *
 -1|     *       *
   |   *           *
 -2| *               *
   |____________________
   -4 -3 -2 -1  0  1  2  3  4
```

This is a rough sketch of the graph. The actual graph would be smoother and continuous.
Graph the function $$ f(x) = -\frac{1}{4} x^{2} $$ as a downward-opening parabola with vertex at (0, 0) and points (2, -1), (-2, -1), (4, -4), (-4, -4).

Graph the following function.

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