D10: Can the graph of a polynomial function ever cross a vertical asymptote? Why or why not? How about crossing a horizontal asymptote? Why or why not? Solution The graph of a function neither crosses the horizontal asymptote not the vertical asymptote. The graph approaches thease asymptotes as x approaches + or - infinity and as y approaches + or - infinity respectively. This is evident from the definition of an asymptote itself. If the graph crosses a line, the concerned line would not be an asymptote at all..