1) Which statement is true about the graph? a) It has a horizontal asymptote of y  = -4 b) It has a vertical asymptote of x = -2 c) Both a) and b) d) None of the above

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lfryerda's profile pic

lfryerda | High School Teacher | (Level 2) Educator

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An asymptote is a line that the graph approaches, but never meets.  

From the graph, we can see that there is a vertical line that the graph approaches at x=-2 but there is no horizontal line that the graph approaches.  The graph is continuing to rise above the horizontal axis on the right side.

This means that the graph has a vertical asymptote at x=-2 and no horizontal asymptote.  Therefore the statement that is true about the graph is part (b).

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embizze's profile pic

embizze | High School Teacher | (Level 2) Educator Emeritus

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The definition of an asymptote is a line that the graph approaches as x grows without bound. There is no reason that the function cannot touch the asymptote. For rational functions this is true, but consider `y=1/x sinx+3` . This function approaches the line y=3 while crossing it an infinite number of times.

I realize that the true definition requires limits, but we must be careful to define the terms accurately, if not precisely.