How do I graph the rational function: #y=6/x+4#?
1 Answer
I like to identify the following things first, when asked to graph a rational function:
 yintercept(s)
 xintercept(s)
 vertical asymptote(s)
 horizontal asymptote(s)

To identify the yintercept(s), ask yourself "what is the value of y when x=0"?
#y = 6/0+4#
Since#6/0# is undefined, there is no yint
yintercept: none 
To identify the xintercept(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#
xintercept:#(3/2,0)# 
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# 
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:
Range: