The techniques of sketching a curve consist of

1. Determining the domain of a curve
2. Finding first and second derivatives (to find relative max, relative minima and point of reflection)
3. Noting any symmetry
4. Finding any asymptotes
5. Noting some special values
6. Considering some limits as x and y go to infinity
7. Construct a table

These techniques are used to determine the curve’s trend.  They may not be applicable in all cases, but, depending on the particular problem, some can always be used.  

Example 1, Example 2, Example3 (Also include GraphFunc applet that is available to be used in sketching the graph online.  See the demo)

Example 1.  Determine the relative maxima, relative minima, and points of inflection of the function:

     .

Sketch the graph.  Use the GraphFunc utility online to confirm the results.

Solution

1.      The domain of f(x) is determined for all.

2.      Derivatives:and .

·          x = 0 => y = 0

·         => y = 9/4

·         => y = -9/4

·         x = 1 => y = -5/4

·         x = -1 => y = -5/4

3.     

4.      Construct a table:

Based on the table’s results, we conclude

·         and are relative minima becausechanges sign from negative to positive as x pass through these points.

·         x = 0 is relative maxima becausechanges sign from positive to negative as x pass through this point.

·         are the points of inflection because(second derivative) changes sign as x pass through these points.

·         The graph’s trend is shown on the constructed table.  The direction of the arrow is down or up in each interval is depended on the sign of is negative or positive, respectively.  The Figure 1 illustrates the graph of f(x).

Figure 1 – the graph of

• Begin by going to http://graph.seriesmathstudy.com. (you need to wait for the GraphFunc applet to be loaded.)

• Enter the expression (x^4)/4 – (3*x^2)/2 at the command line and then press on the Graph It! button to plot the graph.

• To find the critical points, select the Extremum item from the Function dropdown list. Then click on any points near a vertice of the curve to find an extreme point. For instance, the GraphFunc finds the relative minima when being clicked with the mouse at any points near to x =, namely f(1.732) = -2.25, which is illustrated in Figure 2.

Figure 2