Function Name

Parent Function

Graph

Characteristics

Algebra

Constant

Domain:

Range:  or

Inverse Function: Undefined (asymptote)

Restrictions:  is a real number

Odd/Even: Even

General Form: 

Slope Intercept form:

Linear

or

Identity

Domain:

Range:

Inverse Function:

Restrictions:

Odd/Even: Odd (symmetric about the origin)

General Forms:

 or

Quadratic

or

Square

Domain:

Range:

Inverse Function:

Restrictions: None

Odd/Even: Even

Forms:

Standard form:

Vertex form:

Factored Form:

                   Or     

General Form:

Square Root

Domain:

Range:

Inverse Function:

Restrictions:

Odd/Even: Neither

General Form:

Function Name

Parent Function

Graph

Characteristics

Absolute Value

Domain:

Range:

Inverse Function:   

Restrictions:

Odd/Even: Even

General Form:

Cubic

Domain:

Range:

Inverse Function: 

Restrictions: None

Odd/Even: Odd

General Form:

Cube Root

Domain:

Range:

Inverse Function: 

Restrictions: None

Odd/Even: Odd

General Forms:

Exponential

or

Domain:

Range:

Inverse Function:

 or 

Restrictions: None,  can be imaginary

Odd/Even: Neither

General Form:

Logarithmic

or

Domain:

Range:

Inverse Function:

or

Restrictions:

Odd/Even: Neither

General Form:

Function Name

Parent Function

Graph

Characteristics

Reciprocal

or

Rational

Domain:

Range:

Inverse Function:

Restrictions:

Odd/Even: Odd

General Form:

Greatest

Integer

or

Floor

Domain:

Range:  whole numbers only

Inverse Function: Undefined (asymptotic)

Restrictions: Real numbers only

Odd/Even: Neither

General Form:

Inverse Functions

If ,

then

Domain of  Domain of

Range of  Range of

Inverse Function: By definition

Restrictions: None

Odd/Even: Odd

General Forms:

Conic Sections

Circle

Domain:

Range:

Inverse Function: Same as parent

Restrictions: None

Even/Odd: Both

Focus:

General Forms:

The coefficients of  and  are the same sign and same value  

and

Function Name

Parent Function

Graph

Characteristics

Ellipse

Domain:

Range:

Inverse Function:

Restrictions: None

Even/Odd: Both

Foci:  or

General Forms:

and

The coefficients of  and  are the same sign (both positive);

but different values  

and

where

Parabola

Domain:

Range:   or  

Inverse Function:

Restrictions: None

Even/Odd: Even

Vertex:

Focus:

General Forms:

where  and either  or

Hyperbola

Domain:

Range:

Inverse Function:

Restrictions: Domain is restricted

Even/Odd: Both

Foci:   or 

General Forms:

where  or

where

Function Name

Parent Function

Graph

Characteristics

Trigonometry

Sine

Domain:

Range:

Inverse Function:    

Restrictions: None

Even/Odd: Odd

General Form:

Cosine

Domain:

Range:

Inverse Function:    

Restrictions: None

Even/Odd: Even

General Form:

Tangent

Domain:  except for  

Range:

Inverse Function:    

Restrictions: Asymptotes at

Odd/Even: Odd

General Form:

Secant

Domain:  except for

Range:

Inverse Function:    

Restrictions: Range is bounded

Odd/Even: Even

General Form:

Cosecant

Domain:  except for

Range:

Inverse Function:    

Restrictions: Range is bounded

Odd/Even: Odd

General Form:

Cotangent

Domain:  except for

Range:

Inverse Function:    

Restrictions: Asymptotes at

Odd/Even: Odd

General Form:

Function Name

Parent Function

Graph

Characteristics

Arcsine

Domain:

Range:  or Quadrants I & IV

Inverse Function:    

Restrictions: Range & Domain are bounded

Odd/Even: Odd

General Form:

Arccosine

Domain:

Range:  or Quadrants I & II

Inverse Function:    

Restrictions: Range and Domain are bounded

Odd/Even: None

General Form:

Arctangent

Domain:  

Range:  or Quadrants I & IV

Inverse Function:    

Restrictions: Range is bounded

Odd/Even: Odd

General Form:

Arcsecant

Domain:

Range:  or Quadrants I & II

Inverse Function:    

Restrictions: Range & Domain are bounded

Odd/Even: Odd

General Form:

Arccosecant

Domain:

Range:  or Quadrants I & IV

Inverse Function:    

Restrictions: Range & Domain are bounded

Odd/Even: Odd

General Form:

Arccotangent

Domain:

Range:  or Quadrants I & II

Inverse Function:    

Restrictions: Range is bounded

Odd/Even: Neither

General Form:

Function Name

Parent Function

Graph

Characteristics

Hyperbolics

Hyperbolic
Sine

Domain:

Range:

Inverse Function:    

Restrictions: None

Odd/Even: Odd

General Form:

Hyperbolic Cosine

Domain:

Range:

Inverse Function:   

Restrictions: None

Odd/Even: Even

General Form:

Hyperbolic Tangent

Domain:  

Range:

Inverse Function:   

Restrictions: Asymptotes at

Odd/Even: Odd

General Form:

Hyperbolic Secant

Domain:

Range:

Inverse Function:   

Restrictions: Asymptotes at

Odd/Even: Even

General Form:

Hyperbolic Cosecant

Domain:

Range:

Inverse Function:   

Restrictions: Asymptotes at

Odd/Even: Odd

General Form:

Hyperbolic Cotangent

Domain:

Range:

Inverse Function:   

Restrictions: Asymptotes at ,      

Odd/Even: Odd

General Form:

Function Name

Parent Function

Graph

Characteristics

Hyperbolic
Arcsine

Domain:

Range:

Inverse Function:   

Restrictions: None

Odd/Even: Odd

General Form:

Hyperbolic Arccosine

Domain:

Range:

Inverse Function:   

Restrictions:

Odd/Even: Neither

General Form:

Hyperbolic Arctangent

Domain:  

Range:

Inverse Function:   

Restrictions: Asymptotes at

Odd/Even: Odd

General Form:

Hyperbolic Arcsecant

Domain:

Range:

Inverse Function:   

Restrictions:

Odd/Even: Neither

General Form:

Hyperbolic Arccosecant

Domain:

Range:

Inverse Function:   

Restrictions: Asymptotes at           

Odd/Even: Odd

General Form:

Hyperbolic Arccotangent

Domain:

Range:

Inverse Function:   

Restrictions: Asymptotes at ,      

Odd/Even: Odd

General Form:

The Six Functions

“Levers”

Graphing Tips

1)     Move up/down 

k      Vertical translation

“” Moves it up

2)     Move left/right 

h      Phase shift

““ Moves it right

3)     Stretch up/down 

a      Amplitude

Larger stretches it taller or makes it grow faster

4)     Stretch left/right 

b      Frequency ⦁ 2

Larger stretches it wider

5)     Flip about -axis 

a a

If    then odd function

6)     Flip about -axis 

b b

If    then even function

The Six Trig “Levers”

Graphing Tips

Notes

1)     Move up/down 

k      Vertical translation

If  then -axis is replaced by -axis

2)     Move left/right 

h      Phase shift

‘‘ shifts right

3)     Stretch up/down 

a      Amplitude

a is NOT peak-to-peak on  -axis

4)     Stretch left/right 

b      Frequency ⦁ 2

T  peak-to-peak on -axis

  for   

5)     Flip about -axis 

a a

Odd Function:

6)     Flip about -axis 

b b

Even Function: