How do you graph #f(x)=1/(x^2-16)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Given: #f(x) = 1/(x^2 - 16)#

The given function is a rational (fraction) function of the form:

#f(x) = (N(x))/(D(x)) = (a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0)/(b_mx^m + b_(n-1)x^(n-1) + ... + b_1x + b_0)#

First factor both the numerator and denominator. The denominator is the difference of squares: #(a^2 - b^2) = (a + b)(a - b)#

#f(x) = 1/(x^2 - 16) = 1/((x +4)(x - 4))#

Holes are found where linear terms can be cancelled from the numerator and denominator. In the given function, there are no holes .

Vertical asymptotes are found when #D(x) = 0# (Don't include hole factors):

#D(x) = (x +4)(x - 4) = 0; " " **x = -4** , " " **x = 4** #

Horizontal asymptotes are related to #n " and "m#:

#n < m#, horizontal asymptote is #y = 0#
#n = m#, horizontal asymptote is #y = (a_n)/(b_n)#
#n > m#, no horizontal asymptote

For the given function #n = 0, m = 2 => n < m:# horizontal asymptote is y = 0

Find intercepts:

Find #x# - intercepts: set #N(x) = 0#

No x-intercepts

Find #y# - intercepts: set #x = 0#:

#f(0) = 1/(0-16) = -1/16#

#y#-intercept: (0, -1/16)