Graphs of Functions and Equations
I. Points, Lines, Polygons, and Graphs of Explicit Functions
The “pln” operation helps plot points, lines and polygons on the coordinate plane by given pairs of points. The array “pln; pt = (x1, y1)” plots the point (x1, y1) on the plane, where “pt” is the keyword, and (x1, y1) are the coordinates. Plot two or more points by “pln; pt = [(x1, y1), (x2, y2), … ]”. In case of two points, their distance is calculated and displayed on the top of the graph.
Replacing the keyword “pt” with “ln” or “pg”, one can to plot lines and polygons by the "pln" function. The array “pln; ln = [(x1, y1), (x2, y2)]” plots one line through points (x1, y1) and (x2, y2), and “pln; ln = [(x1, y1), (x2, y2)]; ln=[(x3, y3), (x4, y4)]; …” plots two or more lines on the plane. In case of one line, the equation of the line is given on the top of its graph.
The array “pln; pg = [(x1, y1), (x2, y2), (x3, y3)]” plots a triangle with vertices (x1, y1), (x2, y2) and (x3, y3). Adding one or more vertices to the array, one can plot quadrilateral and polygons of five or more vertices.
The “plt” function plots one or more graphs of explicit functions, and “plt; f(x); g(x); h(x)” plots three function graphs on the same coordinate plane, where f(x), g(x) and h(x) are expressions of three distinct functions. Options of “itv = (a, b)” or “pt = [(x1, y1), (x2, y2), (x3, y3), ...]” can be added to the function for interested intervals or points.
Figure 1: Points, Lines, Polygons, and Function Graphs by “pln” and “plt” Operations
II. Plane Curves for Parametric and Implicit Equations
Both functions "pc2" and "imf" produce plan curves for equations. The array “pc2; x(t); y(t)” plots a 2D curve for the parametric equations x = x(t) and y = y(t), and "imf;f(x,y)" produces a graph of an implicit equation f(x, y) = 0.
A specific interval “itv = (a, b)” can be added to the array, so “pc2; x(t); y(t); itv = (a, b)” plots the parametric curve within an interval [a, b] for parameter t, where “itv” is the keyword for interval.
Add interested points and lines to parametric curves by “pc2; x(t); y(t); pt = (x0, y0)”, “pc2; x(t); y(t); pt = [(x0, y0), (x1, y1)…]”, “pc2; x(t); y(t); ln = [(x0, y0), (x1, y1)]”, and “pc2; x(t); y(t); ln = [(x0, y0), (x1, y1)]; ln = [(x2, y2), (x3, y3)]; …”.
The array “imf; f(x, y)” helps graph the implicit equation f(x, y) = 0, where “y” is implicitly defined as a function of “x”, and “f(x, y)” is an implicit function. To change the default interval, add “x1; x2” for the interval of “x” and “y1; y2” for “y” to the end, and the array becomes “imf; f(x, y); x1; x2; y1; y2”.
Replacing the operation “imf” by “cnt”, one can obtain the contour curves for the implicit function f(x, y).
Figure 2: Level Curves, Contour Lines, Curves for Equations by “pc2”, “imf” and “cnt” Operations
III. Graphs of Polar Functions
For polar curves, the array "pol; f(x); g(x); h(x)" plots the curves of three polar functions, where “f(x); g(x); h(x)” are three distinct polar functions, and “x” is an angle variable measured by radians. To change the default interval of “x” to [a, b], use “pol; f(x); g(x); h(x); itv = (a, b)”.
Figure 3: Graphs of Polar Functions by “pol” Operation
IV. Parametric Equations for Space Curves
For parametric equations x = x(t), y = y(t), and z = z(t), the array "pc3; x(t); y(t); z(t)" plots a 3D space curve for the equations x = x(t), y = y(t) and z = z(t), where "x(t); y(t); z(t)" are the coordinate functions. Use the array "pc3; x(t); y(t); z(t); a; b" to change the default interval of the parameter "t" to an interested interval [a, b].
Figure 4: Space Curves for Parametric Equations by the “pc3” Operation
V. Functions and Parametric Equations for Space Surfaces
For the parametric equations x = x(u, v), y = y(u, v) and z = z(u, v) with two parameters u and v, the array "ps3; x(u, v); y(u, v); z(u, v)" plots a 3D space surface for the equations x = x(u, v), y= y(u, v) and z = z(u, v), where "x(u, v); y(u, v); z(u, v)" are the coordinate functions. To change the default parameter intervals for u and v, use "ps3; x(u, v); y(u, v); z(u, v); a; b; c; d", where [a, b] is the interval for "u" and [c, d] is the interval for "v".
For an explicit function z = f(x, y) of two variables, the array "sf3; f(x, y)" plots its surface. To change the default intervals for the two independent variables “x” and “y”, use the array "sf3; f(x, y); x1; x2; y1; y2", where “x1; x2” represent interval [x1, x2] of “x”, and “y1; y2” the interval [y1, y2] of “y”.