Graph the following function. \[ g(x)=\frac{1}{6} x^{2} \]

Step 1 ：Generate a range of x-values.

Step 2 ：Calculate the corresponding y-values using the function \(g(x)=\frac{1}{6} x^{2}\).

Step 3 ：Plot these points on a graph.

Step 4 ：The graph of the function \(g(x)=\frac{1}{6} x^{2}\) is a parabola opening upwards. The vertex of the parabola is at the origin (0,0). As x increases or decreases, \(g(x)\) increases. The graph is symmetric with respect to the y-axis.

Step 5 ：\(\boxed{\text{The graph of the function } g(x)=\frac{1}{6} x^{2} \text{ is a parabola opening upwards.}}\)