How do I graph the function: #y=(2x)/(x^21)#?
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 = (2(0))/((0)^21)=0/1=0#
yintercept: (0,0) 
To identify the xintercept(s), ask yourself "what is the value of x when y=0"?
For this problem, since we've already identified that the graph goes through (0,0), we have both the xint and yint complete! But in case you didn't realize...
#0=(2x)/(x^21)# means that the numerator of the fraction must = 0
#0=2x#
#0=x#
xintercept: (0,0) 
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=(2x)/(x^21)#
#y=(2x)/((x+1)(x1))#
Undefined when denominator = 0:#(x+1)(x1)=0#
Vertical asymptotes:#x=1, x=1# 
To identify the horizontal asymptotes, we think of the limiting behavior (ie: what happens as x gets HUGE)
#y=(2x)/(x^21) > y = "huge" / "HUGER" > 0#
Horizontal asymptote:#y=0#
Now you might pick a couple additional points to the left/right/between your horizontal asymptotes to get a sense of the graph shape.

Pick a point to the left of the
#x=1# asymptote, ie:#x=2#
#y=(2(2))/((2)^21) = 4/(41) = 4/3# Point 1:#(2, 4/3)# 
Pick a point between the two asymptotes We already have the point (0,0) from above. Point 2:
#(0,0)# 
Pick a point to the right of the
#x=1# asymptote, ie:#x=2#
#y=(2(2))/((2)^21) = 4/(41) = 4/3# Point 3:#(2, 4/3)#
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