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Graphing cubic functions, such as the equation x=y^3, involves a few specific steps to accurately depict the function's behavior. Cubic functions typically have an 'S' shaped curve, with one portion of the graph increasing and another decreasing. To start, you would select a range of y-values and calculate the corresponding x-values. For instance, if y=-2, y=1, or y=2, you would find the x-values to be -8, 1, and 8 respectively.

By plotting these pairs, and more, on a graph, one creates a set of points through which the graph's curve can be drawn. Remember to choose enough points and cover a broad range of y-values to accurately reflect the curve's shape. Different cubic equations may demonstrate variations in their curves, but they all follow the basic 'S' shape that is characteristic of cubic equations.

When sketching the graph, pay close attention to how the curve crosses the axes, as this gives insight into intercepts—an important characteristic of the graph.

Intercepts are points where the graph crosses the x and y axes. They provide vital information about the function's behavior and are particularly helpful for visualizing algebraic equations on a graph. For the equation x=y^3, we find the x-intercept by setting y=0, which gives us the point (0,0).

The process to find the y-intercept is similar, except here we set x=0 and solve for y, which, for this particular function, also results in (0,0). It is important to note that some cubic functions may have more than one x-intercept or y-intercept, and intercepts do not necessarily have to be at origin. However, for the equation provided, the single intercept at the origin (0,0) is indicative of both the x and y intercepts.

Testing a graph for symmetry is a reliable way to determine if a function behaves the same on both sides of an axis or through the origin. There are three main types of symmetry to consider: symmetry about the y-axis, symmetry about the x-axis, and symmetry about the origin.

For symmetry about the y-axis, you would test whether the point (-x, y) lies on the graph whenever (x, y) does. With the equation x=y^3, replacing x with -x does not satisfy the original equation, indicating the graph is not symmetric about the y-axis.

Similarly, for x-axis symmetry, one would see if the point (x, -y) is on the graph when (x, y) is. This function fails this test as well, confirming that it does not have x-axis symmetry. Lastly, a graph has origin symmetry if for every point (x, y), there is a corresponding point (-x, -y). This function, as indicated by replacing with negative values, does not fulfill the condition for origin symmetry either. Therefore, it can be concluded that the graph of x=y^3 does not exhibit any of the common symmetries.