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 (can graph multiple equations on desmos)
         b) substitution (using one equation set one variable in terms of the other, the substitute into
               the remaining equation):
                 y = 3x-6           from 2nd equation
                 5x + 2(3x-6) = 7   substitute into first
                 11x - 12 = 7       solve for x, then can find y
         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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