Sketching graph of polynomials

Question

Hi,

can u help explain how to sketch the graph of y=x^3+2x^2-x+1

cuz I thought the maxima, minima and x-intercept is irrational (2.5155.... etc.)

but I did find a graph on it online

Answer 1

First find the derivative set it equal to 0 3x²+4x-1=0

This is a quadratic equation the discriment =5.29 so the roots are distinct and real.

The roots are x=.22 and 1.55 approx

To find out if they are max. or min use second derivative rule evaluate y'' (6x+4) at each root if it's positive then is a min if it's negative a max at x=.22 y'' =-5.3 approx and at x=1.55 y''=5.4 and at y''=0 (x=-.667)is an inflection point.

so now we have enough to sketch.... starting from left to right

1. large -x, y→-∞
2. local max at x=-1.55
3. inflection point at x=-.667
4. local min. at x=.22
5. large x, y→∞

Let me know if you have questions

Jim

Answer 2

It is easier to see the graph using desmos.com which is an online graphing tool.