Expert: Alon Mandes - 10/28/2009

Question
Hello,
I need to sketch a graph of y=x^4 +4x^3
I need to find the horizontal and vertical asymptotes by limit definitions.

For the vertical asymptotes I have
lim x-> a+ = +- infinity
lim x-> a- = +- infinity

With a rational function I set the denominator zero and the x - values are the "a" value I use in the limits, to find out is it is + or - infinity

How can I do that for a polynomial?
Has a polynomial a vertical asymptote at all?

Thanks!  

Answer
Hello Vivian,
All polynomial functions such as
    N
f(x)=∑ (an)x^n  
   n=1
Are continues %26 have no Vertical nor Horizontal asymptotes . It can be rather concave up or
concave down , depending on f''(x) . If f''(x)>0 for a<x<b , then its concave up %26 etc' .
To sketch a function such as y=x^4+4x^3 we need to follow the following steps :
#1 Finding points of intercept with y-axis
  We find y(0) : y(0)=0 .
#2 Finding points of intercept with x-axis
  We solve y=0 : x^4+4x^3=0 --> x^3(x+4)=0 . Solutions x=0 %26 x=-4 .
#3 Finding Extremum points (max %26 min)
  To do so we solve the equation y'=0 : y'=4x^3+12x^2=0 --> x+3=0 --> x=-3 .
  To find if this point is max or min , we need to derive 2nd time :
  y''=12x^2+24x ; y''(-3)=12*(-3)^2+24*(-3)=36 . y''(-3)>0 therefore x=-3 is min .
#4 Finding range of rising %26 decreasing
  Rising range : x<-3 %26 decreasing range : x>-3 .   


Alon.