Step By Step Calculus » 13.7 - Graph Sketching for Functions

Synopsis
The ultimate goal in this section is to describe a graph well enough to be able to sketch it. Using what has been covered in the previous sections, we can define a general step-by-step procedure for curve sketching:
Step 1: (Differentiate twice) Evaluate f'f' and f''f''.
Step 2: (Find the important points) Using f, f', f''f, f', f'', find
• Singularities: The set SS of values of xx for which ff is undefined.
• Zeros: The set ZZ of values of xx for which f(x)=0f(x)=0. These values in ZZ are the xx-intercepts.
• Critical points: The set CC of values of xx for which f'(x)=0f'(x)=0 or f'(x)f'(x) does not exist.
• Possible inflection points: The set II of values of xx for which f''(x)=0f''(x)=0 or f''(x)f''(x) does not exist.
Step 3: (Find the asymptotes) Find horizontal, vertical and slant asymptotes if they exist.
Step 4: (Determine the regional behaviour of ff) Divide \mathbb{R}\cap(S\cup Z\cup C\cup I)^C\equiv \mathbb{R}-(S\cup Z\cup C\cup I)\mathbb{R}\cap(S\cup Z\cup C\cup I)^C\equiv \mathbb{R}-(S\cup Z\cup C\cup I) into regions and find behaviour of ff on each.
Step 5: (Sketch the curve) First identify the important points found in Step 2 and then sketch the curve on each region identified in Step 4 with the corresponding local behaviour. Remember the asymptotes (if any) that may help us to understand the behaviour of ff at some end points of the regions.
Step 6: (Label the curve) Label the axes with values corresponding to the important points. Write the co-ordinates beside the important points.