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All direct variation graphs are linear and they all go through the origin.
Do all linear graphs have proportional relationship
You can't trace this graph without going twice over the same edge. In this graph four vertices have degrees 3 (odd) and one has degree 4 (even). In graphs traceable without lifting pencil off the paper and without going over the same edge twice all degrees must be even (enter and leave), and only two degrees can be odd (leave the starting vertex and enter the ending one). Hope this helps.
Polynomials have graphs that look like graphs of their leading terms because all other changes to polynomial functions only cause transformations of the leading term's graph.