No, they don't.
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Inverse variation does not pass through the origin, however direct variation always passes through the origin.
Assuming both the scales on the graph are linear (that is to say that the numbers go up evenly) then YES, a graph which shows direct proportion must be a straight line. It must also pass through the origin (0,0). A straight line which does not pass through the origin is NOT showing direct proportion. Duncan
For any polynomial equation, there must be no constant term.For equations in general there is no easy way. Substitute 0 for each of the variables and if the resulting expression is 0 then the line passes through the origin.
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Slope = 1Y-intercept = 0Y = X