What are the asymptotes of #y=1/x-2# and how do you graph the function?

Question

Genesis Allen · Accepted Answer

The most useful thing when trying to draw graphs is to test the zeroes of the function to get some points that can guide your sketch. Consider #x = 0#:
#y = 1/x - 2#
Since #x=0# cannot be substituted in directly (since it's in the denominator), we can consider the limit of the function as #x->0#. As #x->0#, #y->\infty#. This tells us that the graph blows up to infinity as we approach the y-axis. Since it will never touch the y-axis, the y-axis is a vertical asymptote. Consider #y = 0#:
#0 = 1/x - 2#
#x= 1/2# So we have identified a point that the graph passes through: #(1/2,0)# Another extreme point we can consider is #x->\infty#. If #x->+\infty#, #y-> -2#. If #x->-\infty#, #y->-2#. So at both ends of the x-axis, y will approach -2. This means there is a horizontal asymptote at #y=-2#. So we have found out the following:
Vertical asymptote at #x=0#.
Horizontal asymptote at #y=-2#.
Point contained in graph: #(1/2,0)#. graph{1/x -2 [-10, 10, -5, 5]} You should notice that all three of these facts provide enough information to draw the graph above.

Hazel Anglin · Answer

undefined The asymptotes of the function y=1/x-2 are a vertical asymptote at x=2 and a horizontal asymptote at y=0. To graph the function, plot points on the graph by choosing different values for x and calculating the corresponding y-values. Connect the points to form a smooth curve.

Genesis Allen · Answer

undefined The asymptotes of the function y = 1/(x - 2) are a vertical asymptote at x = 2 and a horizontal asymptote at y = 0. To graph the function, plot points on either side of the vertical asymptote (x = 2) and observe the behavior of the function as x approaches 2. Similarly, plot points for large positive and negative values of x to observe the behavior of the function as x approaches infinity and negative infinity, respectively. Connect these points smoothly, and the resulting graph will show the curve approaching the vertical and horizontal asymptotes.