  ## Function and Relation Library

This set of pages is an extension of the "Function Library" in the text Exploring Functions through The Use of Manipulatives.

Three Most Important Functions:
identity, x
opposite, -x
reciprocal, 1/x
Lines:
vertical lines, x = a, a = some constant
horizontal lines, y = a, a = some constant
linear functions, y = ax + b, etc.
Equations of Lines, y - y1 =m (x - x1), etc.
Solving Linear Equations Graphically, Solve ax + b = cx + d
Parent Functions, Their Slope Functions, and Area Functions
Linear vs Exponential Growth

Ways New Functions Are Created
piece-wise defined function
composition of functions -- how functions are added, subtracted, multiplied and otherwise composed,
dilation by a constant, not just y=f(x) but y=af(x), where a is a constant,
dilation by a nonconstant function, not just y=f(x) or y=af(x), but y = g(x)f(x) and y = g(x)/f(x)
shift, not just y=f(x), y=f(x)+a or y =f(x) +g(x)

A Family of Quadratics y = ax2 + bx + c
Squaring, y = x2, y = x ·x
Square Root Functions, y = x
Roots and Exponents, x is x1/2

Polynomial Functions or Power Functions, y = Axa, or y = x3, y = x ·x ·x, etc.

Rational Functions, y = 1/x3, y = 1/(x+2), etc.

Exponential Functions, bx
Even More About e, the base of the Natural Logs
Exponential Function, exp(x) or ex
Linear vs Exponential Growth
Parent Functions, Their Slope Functions, and Area Functions
Lograthmic Functions, logb(x)
Natual Log Function, ln(x) or loge(x)

More Examples of Composite Functions:
Absolute Value Function, |x|
Conic Sections:
Circle, x2 + y2 = r2
Ellispe, x2/a2 + y2/b2 = 1
Hyperbola, x2/a2 - y2/b2 = 1
Parabolas - A Family of Quadratics y = ax2 + bx + c

Trigonometric Functions, each function
sine, sin(x)
cosecant, csc(x), 1/sin(x)
cosine, cos(x)
secant, sec(x), 1/cos(x)
tangent, tan(x), sin(x)/cos(x)
cotangent, cot(x), 1/tan(x)

Parent Functions, Their Slope Functions, and Area Functions
This page gives the reader exposure and play time with many functions, their slope functions (derivatives), and their area functions (integrals).

This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).

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