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

- 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 - y
_{1}=m (x - x_{1}), etc. - Solving Linear Equations Graphically, Solve ax + b = cx + d
- 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 = ax
^{2}+ bx + c - Squaring, y = x
^{2}, y = x ·x - Square Root Functions, y = x
- Roots and Exponents,
x is x
^{1/2} - Polynomial Functions, y = x
^{3}, y = x ·x ·x, etc. - Rational Functions, y = 1/x
^{3}, y = 1/(x+2), etc. - Exponential or Power Functions, b
^{x} - A Bit about e
- Exponential Function, exp(x) or e
^{x} - Lograthmic Functions, log
_{b}(x) - Natual Log Function, ln(x) or log
_{e}(x) - More Examples of Composite Functions:
- Absolute Value Function, |x|
- Conic Sections:
- Circle, x
^{2}+ y^{2}= r^{2} - Ellispe, x
^{2}/a^{2}+ y^{2}/b^{2}= 1 - Hyperbola, x
^{2}/a^{2}- y^{2}/b^{2}= 1 - Parabolas - A Family of Quadratics y = ax
^{2}+ 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)

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

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