Question

How do I identify the x-intercept(s) and vertical asymptote(s): `y=(x^3+27)/(3x^2+x)`?

Answer 1

1. To identify the x-intercepts, you want to ask yourself: "where does the graph hit the x-axis, 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 x-intercept: (-3,0)

2. 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`