Functions and Quadratics: Concepts and Techniques

Relation

a set of ordered pairs

Function

defined as a relation between a set of input and permissible outputs with the property that each input is related to at most one output

Domain

set of input values (x) for which a function is defined

Range

set of all possible output values (y or f(x)) as dictated by the function and its domain

Domain restrictions

cannot divide by 0 or square root a negative number

Slope intercept form

y = mx + b

Standard form

ax + by = c, a must be a natural number

Point-slope form

y-y1=m(x-x1)

Parallel lines

have the same slope

Perpendicular lines

have opposite reciprocal slopes

Piecewise function

a function that is defined on multiple intervals

Composite functions

are not commutative

Quadratic function (standard form)

ax²+bx+c=0, where a can't equal 0 and a,b, and c are real numbers

Factored form

y = (x-r1)(x-r2)

Linear difference

same distance between values (the first degree of finite difference)

Quadratic difference

same second distance between values (the second degree of finite difference)

Symbolic form of quadratic functions

y = ax²+bx+c

Graphical form of quadratic functions

parabola, y=x²

Numeric form of quadratic functions

constant second difference

Vertex form

f(x)=a(x-h)²+k —-- (h,k) is the vertex

ZPP (zero product property)

if ab=0, then either a=0 or b=0 or a=b=0

Completing the square

add ½ of b squared as the c value, and then subtract that same value from k

Domain of a quadratic

all real numbers

Finding y-intercepts

by setting all x's equal to 0

Finding x-intercepts

by setting y's equal to 0