Function Families
Polynomial Functions
Function
Parent
equation
Graph
Roots?
Bends
End Behavior
Restrictions?
6th Degree
f (x) = x6
0,1,2,3,4,5,6
Max 5
Starts and finishes in
the same direction;
leading coefficient is
positive – up/up;
leading coefficient is
negative – down,
down
None on domain;
Range restricted
Linear
f (x) = x
One root
None
Starts and ends in
opposite directions;
slope is positive –
starts down, finishes
up; slope is negative
– starts up and
finishes down
None
Quadratic
f (x) = x2
0, 1 or 2
1
Starts and finishes in
the same direction;
leading coefficient is
positive – up/up;
leading coefficient is
negative – down,
down
None on domain;
Range restricted
Function
Parent
equation
Graph
Roots?
Bends
End Behavior
Restrictions?
Cubic
f (x ) = x3
1, 2, or 3
Maximum 2
Starts up and finishes
down; leading
coefficient positive –
down/up; leading
coefficient negative –
up/down
None
Quartic
f (x) = x4
0. 1. 2. 3 or 4
Maximum 3
Same direction;
positive coefficient –
up/up; negative
coefficient –
down/down
None on domain;
Restriction on
range
5th Degree
f (x ) = x5
1, 2, 3, 4 or 5
Maximum 4
Opposite directions;
Positive coefficient –
down/up; Negative
coefficient – up/down
None
Function
Parent
equation
Graph
Roots?
Bends
Odd Degree
Minimum of 1;
maximum =
degree
Maximum =
degree – 1
Even Degree
Maximum =
degree; May
have 0
Maximum =
degree – 1
End Behavior
Opposite directions;
Positive coefficient –
down/up; Negative
coefficient – up/down
Same direction;
positive coefficient –
up/up; negative
coefficient –
down/down
Restrictions?
None
None on domain;
Range restricted
Function Families
Special Functions
Function
Rational
Function
Parent equation
f ( x) =
1
x
Graph
Definition
Special Features
p( x )
where
q( x )
p( x ) and q ( x ) are polynomial
An equation of the form
functions, and q ( x ) ≠ 0 .
Square Root
Function
f (x) = x
Restrictions?
Domain: Look at
restrictions on the
denominator;
denominator cannot
equal 0
CALCULATOR:
Set to DOT mode
Range: Look at
numerator – can it
equal 1 or 0?
A function that contains a square root of
a variable.
Domain: expression
under the radical
cannot be negative
Range: will be positive
unless shifted on the
graph
Function
Absolute
Value
Function
Parent equation
f (x) = x
Graph
Definition
Special Features
A function written as f ( x ) = x , where
Domain: no restriction
f ( x ) ≥ 0 for all values of x .
Range: absolute value
is positive unless
shifted on the graph
CALCULATOR:
MATH
NUM
1. abs
Piecewise
Function
Example:
−2x + 3;
f (x) =  2
x ;
x < −1
x ≥ −1
Restrictions?
A function that is written using two or
more expressions.
CALCULATOR:
Set to DOT mode
Use TEST menu
y 1 = ( −2 x + 3 )( x < −1)
y 2 = ( x 2 )( x ≥ −1)
Depends on the
functions in the
equation.
Function
Greatest
Integer
Function
Parent equation
f (x) = x
Graph
Definition
Special Features
A step function, written as f ( x ) = x ,
where f ( x ) is the greatest integer less
than or equal to x .
CALCULATOR:
MATH
NUM
5. int
Restrictions?
Domain: all real
numbers
Range: integers only