The study of functions, graphing functions

Example.

1. Scope





2. Parity, odd parity, periodicity

The function is neither even nor odd and periodic

3. The intersection with the coordinate axes





or

4. The derivative and critical points



or

5. Intervals of increase, decrease, and extreme points (and the value of the function at these points)




6. The behavior of the functions at the ends of the scope and the asymptotes of the graph of a function (vertical, horizontal and inclined)
   

   When you left

   

   In the case

   

   Therefore, — vertical asymptote

   Since

   when then ie is the oblique asymptote

7. The second derivative and the study of the function of the bulge and gnutt

Find the inflection points (if they exist) and values in the points of inflection





So the sign of the second derivative can change only at the point


8. If you want to find control points specifying the behavior of the graph

9. On the basis of the study building sketch the graph of a function