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

Question

Nathan Axel · Accepted Answer

Vertical asymptote: #x=2# and horizontal asymptote: # y=0#
Graph - Rectangular hyperbola as below.

#y = 1/(x-2)# #y# is defined for #x in (-oo,2) uu (2,+oo)# Consider #lim_(x->2^+) y =+oo# And #lim_(x->2^-) y =-oo# Hence, #y# has a vertical asymptote #x=2# Now, consider #lim_(x->oo) y =0# Hence, #y# has a horizontal asymptote #y=0# #y# is a rectangular hyperbola with the graph below. graph{1/(x-2) [-10, 10, -5, 5]}

Abigail Allen · Answer

undefined The function y=1/(x-2) has two asymptotes: 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 and observe the behavior of the function as x approaches positive and negative infinity. Additionally, plot the point (2,0) on the graph.

Nathan Axel · Answer

undefined The asymptotes of $ y = \frac{1}{x-2} $ are the vertical and horizontal asymptotes.

Vertical asymptote: $ x = 2 $
Horizontal asymptote: $ y = 0 $

To graph the function:

1. Plot the vertical asymptote at $ x = 2 $.
2. Determine the behavior of the function as $ x $ approaches positive and negative infinity to sketch the horizontal asymptote at $ y = 0 $.
3. Choose several points on either side of the vertical asymptote and calculate their corresponding $ y $ values to plot points.
4. Sketch the curve passing through the plotted points, making sure it approaches the asymptotes as $ x $ approaches positive and negative infinity.