The Mathematics of Wallpaper

CONTENTS OF CURRICULUM UNIT 10.04.07

  1. Unit Guide
  1. Why am I writing this unit?
  2. Relations
  3. Functions
  4. The Vertical Line Test
  5. Inverse Relations and Inverse Functions
  6. Restricting the Domain to get Inverse Functions
  7. Applying Translations and Reflections
  8. Music and Mathematics
  9. A reflection in a pentagram
  10. The Unit Circle, the Symmetries of a Square and Dancing
  11. From words to action
  12. Planning for Algebra–II
  13. The second lesson of the curriculum unit
  14. Planning for Trigonometry
  15. References

Patterns, a Different Point of View

Luis Alberto Magallanes

Published September 2010

Tools for this Unit:

The Vertical Line Test

The vertical line test is a simple way of checking if a numerical relation is or is not a function by looking at its graph. If R is a numerical relation, then for a given number x in the domain of R, the values y such that xRy are recorded in the graph of R as points (x,y). For the given x, all the related values of y give points (x, y) that lie on the vertical line through (x, 0). If R is a function, there should be only one such y, and therefore the graph of R will intersect each vertical line in at most one point. In other words, on a graph of a relation, if a vertical line passes through more than one point, the relation is not a function.

image 10.04.07.04

The two relations whose graphs are shown here in red color do not represent functions because the blue vertical line is touching the graph in more than one point.

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