Vertical Asymptote at #x=4#
We can see that there is a vertical asymptote at #x=4#.
When #x# is just slightly less than 4 (eg. 3.9999),
#x-4# is a very small negative number which makes
#1/(x-4)# a very large negative number which makes
#-1/(x-4)# a very large positive number which makes
#y# a very large positive number.
When #x# is just slightly more than 4 (eg. 4.0001),
#x-4# is a very small positive number which makes
#1/(x-4)# a very large positive number which makes
#-1/(x-4)# a very large negative number which makes
#y# a very large negative number.
End Behavior
When #x# is largely negative (eg. #-10^10#), #-3/(x-4)# becomes very close to 0, so #y# is close to #-1#.
When #x# is largely positive (eg. #10^10#), #-3/(x-4)# becomes very close to 0, so #y# is close to #-1#.
So the plot approaches a horizontal asymptote at #y=-1# as its end behavior.
#y#-intercept
When #x#=0, #y=-3/(0-4)-1=-1/4#.
There is a #y#-intercept at #(0, -1/4)#
#x#-intercept
When #y#=0,
#0=-3/(x-4)-1#.
Solve for #x# to get the #x#-intercept.
#x-4=-3#
#x=1#
There is an #x#-intercept at #(1, 0)#
graph{-3/(x-4)-1 [-5, 15, -20, 20]}
