The Ungerecht Family Tree

• Graphing circles, ellipses, parabolas, hyperbolas
• Polynomials, quadratic formula and completing the square

quadratic formula: x = (-b +- √(b²-4ac)) / 2a

2 equations, for 2 unknowns (need n equations to solve for n unknowns)

Example: 5x + 2y = 7, 3x - y = 6
Solve by a) graphing both equations and see where they intersect,
         b) substitution (using one equation set one variable in terms of the other, 
               then substitute into the other equation):
                 y = 3x-6
                 5x + 2(3x-6) = 7
                 11x - 12 = 7
         c) addition - put all like terms in columns, 
                       multiply/divide as needed to get one set of terms to cancel
                       add the terms, solve for the remaining term
               5x + 2y = 7
               6x - 2y = 12
               --------------
               11x     = 19, solve for x, then put in to find y 

Define x and y in terms of t (time). Arrows on graph represent increasing values of t. Allows you to create functions, using two graphs, from things are not functions for both x and y together (circles, ellipses, x = y², etc)

Eliminating the parameter: substitute for t and define x in terms of y (or vice versa), aka rectangular equation.

https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/

• Paul Dawkins notes on Algebra and other topics at https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx
• Math symbols: https://www.rapidtables.com/math/symbols/Set_Symbols.html
• Online graphing calculator: https://www.desmos.com/calculator

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