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 x^1/2

 

Polynomial Functions or Power Functions, y = Ax^a, or y = x³, y = x ·x ·x, etc.

 

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

 

Exponential Functions, b^x

A Bit about e

Even More About e, the base of the Natural Logs    

Exponential Function, exp(x) or e^x

Linear vs Exponential Growth

Parent Functions, Their Slope Functions, and Area Functions

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)

 

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).

© 2008, 2018, 2020 Agnes Azzolino www.mathnstuff.com/math/spoken/here/2class/300/fx/library/library.htm