OVERVIEW OF GRAPHING TECHNIQUES

Graphmatica offers the following methods for graphing equations. Each method is detected automatically by the use of distinctive variables.

Normal Cartesian (rectangular) Functions Typical graphs like y=x^2 including only the variables x and y. Also includes relations like x^2 + y^2 = 36.

Implicit Cartesian Functions Implicit functions are those in which neither x nor y can be isolated as the dependent variable. These are only supported for Cartesian coordinates. Includes relations like x^2-xy+y^2=10.

Inequalities Most Cartesian equations can be graphed as inequalities as well by replacing = with <, <=, >, or >=.

Example: y < x^2

Single Points To graph a single point, just specify the x and y coordinates as shown below.

Example: x = 4; y = 2 plots (4,2).

Data Plots Enter a set of (x,y) coordinate pairs, and optionally find the best-fit curve to predict the data.

Polar Graphs Graphs using the polar coordinate system and the variables r and t (for theta).

Example: r=cos t

Parametric Graphs Graphs using the rectangular coordinate system but specified by equations of a third variable or "parameter," t. These graphs must include a domain.

Example: y = sin t ; x = cos t {0, 2p} draws a circle.

Differential Equations Approximate numerical solutions to differential equations; use variables dx (for the differential dx/dt), x, and t, or dy, y, and x. You can draw slope fields for first-order equations, and initial value approximations for first-order up to fourth order ODEs.

Example: dx = x^2 + t plots a slope field for "dx/dt = x^2 + t."

Example: d2y + y = 0 {0,0,1} plots a sine wave, the solution to "d²y/dt² + y = 0" with initial values y=0 and dy/dx =1 at x = 0.

Systems of ODEs Approximate numerical solutions to linear systems of differential equations, up to fourth order; use variables dx,dy,dz,dw (for the differentials dx/dt, etc.), x,y,z,w, and t, or x1...x4 and t.

Example: dx = 3t-y ; dy = y-x {0,0,1} plots the solutions x(t) and y(t) for initial values x=0 and y=1 at t=0.

Graphmatica comes with pre-defined equation lists demonstrating each of these graph types. See Demo Files for details.

While some curves can be drawn by Cartesian relations, polar coordinates, and parametric functions, each technique is better suited for some graphs than for others. For instance, a circle with radius 5 around the origin which can be produced by the equation

x^2 + y^2 = 25

can be drawn faster by the parametric equations

x = 5 cos (t) ; y = 5 sin (t) {0, 2p}

and can be drawn faster and much more simply by the polar graph

r = 5.