Question 1033529
equation is y = (9x^3 + 2x) / (3x^2 + 1)


if the degree of the numerator is less than or equal to the degree of the denominator, than you will get a horizontal asymptote.


if the degree of the numerator is greater than the degree of the denominator by 1 degree, then you will get a slant asymptote.


the degree of the numerator is one more than the degree of the denominator so you have a slant asymptote.


divide the denominator into the numerator and you will get y = 3x plus a remainder.


forget the remainder and the equation of your slant asymptote is y = 3x.


here's the graph of your equation and the equation of the slant asymptote.


<img src = "http://theo.x10hosting.com/2016/050801.jpg" alt="$$$" </>


the graph of the horizontal or slant asymptote CAN cross the graph of the equation.


that is exactly what it does here.


it crosses the graph of the equation at x = 0.


here's the same graph blown up so you can see the intersection point more clearly.


<img src = "http://theo.x10hosting.com/2016/050802.jpg" alt="$$$" </>


here's a reference on horizontal and slant asymptotes.


<a href = "http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/18-rational-functions-finding-horizontal-slant-asymptotes-01" target = "_blank">http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/18-rational-functions-finding-horizontal-slant-asymptotes-01</a>


you, of course, need to know how to do polynomial division.


if you have a problem with that, look here:


<a href = "http://www.purplemath.com/modules/polydiv2.htm" target = "_blank">http://www.purplemath.com/modules/polydiv2.htm</a.