Introduction to GrafEq

GrafEq (“GrafEq” has the same pronounciation as “graphic”.) is designed for two kinds of mathematicians: the teacher and the student. Teachers will appreciate features such as the variable font size and the ability to cut and paste expressions and graphs to text documents; students will appreciate the standard mathematics notation and ease of use of the program. Everyone will appreciate GrafEq’s mathematical prowess with its broad range of relations that can be displayed with confidence.

GrafEq is an exceptional tool for exploring mathematics. As an introduction, here are some characteristics underlying the design of GrafEq:

- Mathematical
  Algebraic specifications are accepted in a broad variety of intuitive formats. For instance, a linear equation can be entered in the form y = m x + b, (y-k)/(x-l) = m, or a x + b y + c = 0.
- Honest
  Any inaccuracy is explicitly revealed. For instance, although a pixel is very small, it is still an area rather than a point; therefore, the ‘coordinates’ of a pixel are reported using a range of values, such as (5.210±0.002, 17.001±0.002).
- Valid
  Graphs plotted are correctly. For instance, the graph of y<=sqrt(16-x2) differs from the graph of y>!sqrt(16-x2), because ‘<=’ is not equivalent to ‘>!’.
- Reliable
  No solutions will be missed: A pixel is off only if it contains no solutions at all; it stays on if it may possibly contain one or more solutions for the graph’s active relations. Extraneous solutions are eliminated when graphing is complete. The program can even use multiple colours to distinguish the pixels that have been proven to contain solutions, the pixels that have been proven to not contain solutions, and the pixels that are currently undecided.

  Hint: | A closed and open version of a relation will often have identical graphs on the computer’s monitor. For example: the graph of x<5 will be identical to that of x<=5 unless the edge x=5 falls right between two adjacent pixels.

- Multi-faceted
  Each GrafEq graph can be presented in four forms:
  - Symbolic - an algebraic definition, typeset using mathematics notation, and presented in one or more algebraic relation windows. (Each relation can contain one or more constraints, and each constraint can be an equation, an inequality, a set description, or a conditional definition.)
  - Structural - a flow chart, or tree interpretation, presented in one or more structural relation windows. (Each relation is associated with an algebraic window and a structural window.)
  - Graphical - a cartesian or polar representation, presented in one or more view windows.
  - Printed - physical copies can be printed directly from GrafEq by means of its page editing window, or indirectly from other programs such as word processors or drawing applications by first copying the relations and/or graph views from GrafEq, and then pasting into the other program.

This manual provides a convenient and complete reference to the various features and some anticipated application of such features. Various tutorials, from beginner to expert levels, are also useful tools and a starting point for a rewarding journey of mathematics exploration.

Hint: | This manual presumes that the reader will have access to general information about the computer and its operating system, as well as mathematical terminology and concepts. If the illustrations in this manual do not match your display, check the settings in the File / Preference manual. The screen shots are from a computer running Windows 98. GrafEq will run on a black and white display, in which case some references to colours should instead be references to patterns.

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