Identify all holes, asymptots, x & y intercepts with a sketch: Parobolic asymptote 2x^3-3x^2-8x-3/x-2 = (x+1)(2x^2-5x-3)/x-2 (this is 1 equation)

Expert Answers
rcmath eNotes educator| Certified Educator

The right hand side of your equation is the factored form of the left hand side. in other words, what you wrote is y=y.

So I assuming that you are asking us to study `y=(2x^3-3x^2-8x-3)/(x-2)`

1) y-intercepts

x=0 => `y=(-3)/(-2)=3/2`

Hence y-int (0,3/2)

2) x-intercepts

y=0 =>`2x^-3x^2-8x-3=0=>(x+1)(2x^2-5x-3)=0=>`

`(x+1)(x-3)(2x+1)=0=> x=-1, or x=3, or x=-1/2`

Hence x-intercepts are (-1,0), (3,0), (-1/2,0)

3) vertical Asymptote

x=2 makes denominator zero=> horizontal asymptote

x=2 vertical asym

4) No oblique asymptote or horizontal

5)Holes are points that sets the numerator and denominator to be zero, we don't have any in our case.

The following graph confirm our findings.