How to find the asymptotes of #y=22/(x+13)-10#?

Question

Theodore Amidon · Accepted Answer

#"vertical asymptote at "x=-13#
#"horizontal asymptote at "y=-10# The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote. #"solve "x+13=0rArrx=-13" is the asymptote"# Horizontal asymptotes occur as #lim_(xto+-oo),ytoc" ( a constant)"# divide terms on numerator/denominator by x #y=(22/x)/(x/x+13/x)-10=(22/x)/(1+13/x)-10# as #xto+-oo,yto0/(1+0)-10# #rArry=-10" is the asymptote"# graph{((22)/(x+13))-10 [-40, 40, -20, 20]}

Elijah Allen · Answer

undefined To find the asymptotes of $ y = \frac{22}{x + 13} - 10 $, follow these steps:

1. Determine the vertical asymptotes by identifying the values of $ x $ that make the denominator equal to zero.
2. Determine the horizontal asymptote by observing the behavior of the function as $ x $ approaches positive or negative infinity.

For the given function:

1. Vertical asymptote:
   Set the denominator $ x + 13 $ equal to zero and solve for $ x $.
   $ x + 13 = 0 $
   $ x = -13 $

Therefore, there is a vertical asymptote at $ x = -13 $.

2. Horizontal asymptote:
   As $ x $ approaches positive or negative infinity, the function approaches its horizontal asymptote.
   Since the highest power of $ x $ in the numerator and denominator is 1, the horizontal asymptote is determined by the ratio of the leading coefficients.
   
   The horizontal asymptote is given by:
   \( y = \frac{22}{1