Sketching graphs of polynomial functions are useful in graphing techniques, and solving inequalities.
1. First mark down the roots of the polynomial function.
2. Decide how the tail ends behave, whether it is above or below the X axis. This can be determined from the coef of the highest power. For instance, if the polynomial is of degree 6 and the coef of the highest power is positive, then when X approaches either positive or negative infinity, the function approaches positive infinity, so the tail ends are above the X axis.
3. Determine how the function behaves at the roots. When there are 3 or more odd number of roots at the same point, there is a point of inflexion at the root. When there are 2 or more even number of roots at the same point, there is a minimum or maximum point at the root.