"Why = a(x - h)² + k?" -- EVERYTHING ... ABOUT A QUADRATIC ...       Using Quadratics to Create Other Functions

 Quadratics Used to Build Functions     Quadratics are used to build other functions.     To build functions is to compose functions.     To build functions by multiplication is to use dilation.     When quadratic are multiplied or divided, to compose a function, the result is a polynomial or a rational function.

 Techniques of Algebra Work With Functions     Recall the following from algebra I and see how these laws create new functions. (xa)(xb) = xa+b -- to multiply, add exponents   xa ÷ xb = xa-b -- to divide, subtract exponents     Multiplication produces these new functions. (constant)(constant) = (constant) Ex. (5)(2) = 10   (constant)(line) = (line) Ex. (5)(2x+3) = 10x+15   (line)(line) = (quadratic) Ex. (5x)(-x+3) = -5x²+15x   (line)(quadratic) = (cubic) Ex. (5x)(x²+3x+1) = -5x3+15x2+5x   (3rd degree function)(5th degree function) = (8th degree equation) Ex. (5x3)(2x5) = 10x8     Division produces these new functions. (constant)/(constant) = (constant) Ex. (5)/(2) = 2.5   (constant)/(line) = (reciprocal function) , also a rational function Ex. (5)/(2x+3) = (5)/(2x+3)   (line)/(line) = (reciprocal function), also a rational function Ex. (5x)/(-x+3) = 15/(-x+3) - 5   (line)(quadratic) = (rational function) Ex. (5x)/(x²+3x+1)   (quadratic)/(line) = (rational function) Ex. (x²+3x+1)/(5x)   (3rd degree function)/(5th degree function) = (rational function) Ex. (5x3)/(2x5) = 5/2x2       When using multiplication to compose a function, a zero of a factor is a zero of the composed product function.   a restriction in a factor is a restriction of the composed product function.     When using division to compose a function, almost always, a zero of the numerator is a zero of the composed rational function.   almost always, a zero of the numerator creates a vertical asymptote of the rational function.   if the same zero exists in a numerator factor and a denominator factor, a discontinuity (a hole in the function) is created.