$$ y=\frac{1}{3} x^{2} \quad $$ Function

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The function given is $$ y = \frac{1}{3} x^{2} $$. This is a quadratic function, which can be analyzed for various features such as its vertex, axis of symmetry, and intercepts.

Let's analyze the function step by step:

1. **Identify the type of function**: This is a quadratic function in the standard form $$ y = ax^2 + bx + c $$, where $$ a = \frac{1}{3} $$, $$ b = 0 $$, and $$ c = 0 $$.

2. **Vertex**: The vertex of a quadratic function in the form $$ y = ax^2 + bx + c $$ can be found using the formula:
   $$
   x = -\frac{b}{2a}
   $$
   Substituting the values of $$ a $$ and $$ b $$:
   $$
   x = -\frac{0}{2 \cdot \frac{1}{3}} = 0
   $$
   To find the $$ y $$-coordinate of the vertex, substitute $$ x = 0 $$ back into the function:
   $$
   y = \frac{1}{3}(0)^2 = 0
   $$
   Thus, the vertex is at the point $$ (0, 0) $$.

3. **Axis of symmetry**: The axis of symmetry for a quadratic function is the vertical line that passes through the vertex. Therefore, the axis of symmetry is:
   $$
   x = 0
   $$

4. **Intercepts**:
   - **Y-intercept**: This occurs when $$ x = 0 $$:
     $$
     y = \frac{1}{3}(0)^2 = 0
     $$
     So, the y-intercept is at $$ (0, 0) $$.
   - **X-intercepts**: These occur when $$ y = 0 $$:
     $$
     0 = \frac{1}{3} x^{2}
     $$
     This gives $$ x = 0 $$ as the only x-intercept, which is also at $$ (0, 0) $$.

5. **Direction of the parabola**: Since $$ a = \frac{1}{3} > 0 $$, the parabola opens upwards.

6. **Range**: The range of the function is $$ y \geq 0 $$ since the vertex is the minimum point.

In summary, the function $$ y = \frac{1}{3} x^{2} $$ has the following features:
- Vertex: $$ (0, 0) $$
- Axis of symmetry: $$ x = 0 $$
- Y-intercept: $$ (0, 0) $$
- X-intercept: $$ (0, 0) $$
- Opens upwards
- Range: $$ y \geq 0 $$
The function $$ y = \frac{1}{3} x^{2} $$ is a quadratic function with its vertex at $$ (0, 0) $$, axis of symmetry along the y-axis, and it opens upwards. It has only one intercept at the origin $$ (0, 0) $$.

\( y=\frac{1}{3} x^{2} \quad \) Function

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