How do I identify the xintercept(s) and vertical asymptote(s): #y=(x^3+27)/(3x^2+x)#?
1 Answer
Feb 26, 2015

To identify the xintercepts, you want to ask yourself: "where does the graph hit the xaxis, aka: what is x when y=0?"
So let y = 0 and solve for x:
#0=(x^3+27)/(3x^2+x)#
In order for this fraction to equal 0, the numerator of the fraction must equal 0 (remember: denominator = 0 > undefined)
#0=(x^3+27)#
#x^3=27#
#x=3#
So the xintercept: (3,0) 
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=(x^3+27)/(3x^2+x)# is already simplified
Undefined when denominator = 0:#(3x^2+x)=0#
#x(3x+1)=0#
#x=0, 3x+1=0#
Vertical asymptotes:#x=0, x=1/3#