Equations

The use of this tool is illustrated in the examples Trigonometric Functions, Families of Curves,  Polar Coordinates and The Derivative.

The configuration panel of EQUATIONS may have as many lines as desired. Each line defines an equation;  actually it defines the equation's graph. The configuration lines look like these:

y=A*sin(x):colour=red
y=a*x^2+2*b*x+c:colour=black:visible=false
x=exp(s*y):sequence=s[0,1]5
a*x^2+2*b*x*y+c*y^2=1:colour=black:editable=true

In general each line contains an equation in the two variables x and y and it specifies some properties of the equation's graph. All specifications must be separated from all others and from the equation by : . The colour of the graph is specified by the expression colour=red. The default colour is blue.

Descartes draws the graphs of all the equations defned it the configuration panel of EQUATIONS. Below, a brief explanation is given of how this drawings are performed. Descartes also shows the equations inside a text field. If the user wants to hide the equation then it suffices to write visible=false. This will hide the equation, not the graph. An equation's graph is always visible. If it is desired that the equation be modified by the user, then the expression editable=true must be added. In this case the user can modify the equation and, after typing <return>, Descartes will draw the graph of the modified equation.

Nippe Descartes distinguishes three types of equations: y=f(x), x=g(y) and f(x,y)=g(x,y). The graphs corresponding to these three types are drawn by different procedures.

For the equations of the type y=f(x) the applet calculates the values of  f(x) for all  x corresponding to the pixels in the horizontal interval which is visible on the applet, and then joins the consecutive points [x,f(x)] with straight line segments.

For the equations of the type x=g(y) the applet calculates the values of  g(y) for all  y corresponding to the pixels in the vertical interval which is visible on the applet, and then joins the consecutive points [g(y),y)] with straight line segments.

Finally, the equations of the form f(x,y)=g(x,y) are converted into the form F(x,y)=0 by defining  F(x,y)=f(x,y)-g(x,y). In order to find the graph of this type of equation, Decartes uses an adaptation to functions of two variables of Newton's method for finding the zeroes of a function of one variable. Once a point of the graph is found (following the negative gradient), Descartes searches the neighbouring points of the graph by following the level curve through that point, that is, the perpendicular to the gradient. All this is done using several points on the plane as starting points for the process, so Descartes can find the various branches of a graph. If the graph has many branches, then Descartes may not be able to find them all. However, the system works quite well in most of the situations that appear in secondary and higher education.

The following example shows three graphs, one of each type. Observe that the third one has several branches. All three equations are editable so the user can change them by writing on the text fields.

By reducing the scale to 10 or 5, the reader will find one those rare situations in which Descartes misses some of the branches of a graph. The responsibility to avoid these situations is left to the author of mathematics Web pages.

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	Ministerio de Educación, Cultura y Deporte. Aņo 2000