Graphing the Difference

        On my previous post, "Approximation of Points of Intersection," I dealt with the use of finding the points of intersection of two graphs to plot the function h(x), where h(x) = f(x) - g(x).  I have decided now to include this method in a seperate post in order to expand upon the process for the general function h(x), where h(x) = f(x) + g(x).

Original

         The principle of h(x) representing the signed difference between f(x) and g(x) can also be applied to graphing.  Rather than adding the ordinates of the graphs of f(x) and -g(x), simply find the difference, if f(x) is above g(x) then h(x) is positive, otherwise h(x) is negative.

          For example graphing x³ - x², first graph x³ and x² as normal.

Fig. 14 Graph of x³ and x²

           Now, at x = 0 and x = 1 (the point of intersection), f(x) is equal to g(x) and thus h(x)has roots at 0 and 1.  Furthermore, we can observe that between 0 and 1, the distance grows and decreases slightly, below 0 the distance increases dramatically, and above 1 the distance increases at a slightly lesser rate.  Graphing this information we get.

Fig. 15 Graph of x³ - x²

End Of Original

           We can further apply this principle by representing f(x) + g(x) as f(x) - (-g(x)).  As an extension of the original example:

          To graph h(x) = f(x) + g(x), first graph x³ and -x² as normal, then use the process described above to obtain the graph for h(x).  From this we see that the graph is a mirror image of that of the original example.