How do I graph the rational function: #y=-6/x+4#?

I like to identify the following things first, when asked to graph a rational function:
- y-intercept(s)
- x-intercept(s)
- vertical asymptote(s)
- horizontal asymptote(s)

1. To identify the y-intercept(s), ask yourself "what is the value of y when x=0"?
   #y = -6/0+4#
   Since #6/0# is undefined, there is no y-int
   y-intercept: none

2. To identify the x-intercept(s), ask yourself "what is the value of x when y=0"?
   #0 = -6/x+4#
   #-4 = -6/x#
   #-4x = -6#
   #x = -6/-4 = 3/2#
   x-intercept: #(3/2,0)#

3. To identify the vertical asymptotes, we first try and simplify the function as much as possible and then look at where it is undefined
   #y = -6/x+4# is already simplified
   Undefined when #x=0#
   Vertical asymptotes: #x=0#

4. To identify the horizontal asymptotes, we think of the limiting behavior (ie: what happens as x gets HUGE)
   #y = -6/"HUGE" +4 -> 0 + 4 -> 4#
   Horizontal asymptote: #y=4#

Now you might pick a couple additional points to the left/right of your horizontal asymptote to get a sense of the graph shape.

• Pick a point to the left of the #x=0# asymptote, ie: #x=-6#
  #y = -6/6 + 4 = -1 + 4 = 3#
  Point 1: #(−6,3)#
• Pick a point to the right of the #x=0# asymptote, ie: #x=6#
  #y = 6/6 + 4 = 1 + 4 = 5#
  Point 2: #(6,5)#

Domain: #(-oo,0),(0,oo)#
Range: #(-oo,4),(4,oo)#