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

 

Ways New Functions Are Created

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 x^1/2

Polynomial Functions, y = x³, y = x ·x ·x, etc.

Rational Functions, y = 1/x³, 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, 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)

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

© 2007, Agnes Azzolino www.mathnstuff.com/math/spoken/here/2class/300/fx/library/library.htm