Function Parent Graph Characteristics Name Function

[Pages:9]Harold's Parent Functions "Cheat Sheet"

6 November 2019

Function Name

Algebra

Parent Function

Constant

() =

Linear or Identity

() =

Quadratic or Square

() = 2

Square Root

() =

Graph

Characteristics

Domain: (-, ) Range: [c, c] Inverse Function: Undefined (asymptote) Restrictions: c is a real number Odd/Even: Even General Form:

+ = 0

Domain: (-, ) Range: (-, ) Inverse Function: Same as parent Restrictions: m 0 Odd/Even: Odd General Forms:

+ + = 0 = +

- 0 = ( - 0)

Domain: (-, ) Range: [0, ) Inverse Function:

-1 () = Restrictions: None Odd/Even: Even General Form:

2 + + + = 0

Domain: [0, ) Range: [0, ) Inverse Function:

-1 () = x2 Restrictions: 0 Odd/Even: Neither General Form:

() = ( - ) +

Copyright ? 2011-2019 by Harold Toomey, WyzAnt Tutor

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Function Parent Name Function

Cubic

() = 3

Graph

Cube Root

() = 3

Reciprocal or Rational

1 () =

Transcendentals

Exponential

() = 10

() =

Logarithmic

() = log

() = ln

Copyright ? 2011-2019 by Harold A. Toomey, WyzAnt Tutor

Characteristics

Domain: (-, )

Range: (-, )

Inverse Function:

-1 () = 3

Restrictions: None

Odd/Even: Odd

General Form: () = (( - ))3 +

Domain: (-, )

Range: (-, )

Inverse Function:

-1 () = 3

Restrictions: None

Odd/Even: Odd

General Form:

() = 3( - ) +

Domain: (-, 0) (0, )

Range: (-, 0) (0, )

Inverse Function: Same as parent

Restrictions: x 0

Odd/Even: Odd

General Form:

() =

( - ) +

Domain: (-, ) Range: (0, ) Inverse Function:

-1 () = log

-1 () = ln Restrictions: None, x can be complex Odd/Even: Neither General Form:

() = 10((-)) +

Domain: (0, ) Range: (-, ) Inverse Function:

-1 () = 10

-1 () = Restrictions: x > 0 Odd/Even: Neither General Form:

() = log(( - )) +

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Function Parent Name Function

Absolute Value

() = ||

Greatest Integer or Floor

() = []

Graph

Characteristics

Domain: (-, )

Range: [0, )

Inverse Function:

-1 () = 0

Restrictions:

()

=

{-,,

0 < 0

Odd/Even: Even

General Form:

() = |( - )| +

Domain: (-, ) Range: (-, ) whole numbers only Inverse Function: Undefined (asymptotic) Restrictions: Real numbers only Odd/Even: Neither General Form:

() = [( - )] +

Inverse Functions

(-1 ()) = -1 (()) =

Domain of x Domain of y Range of y Range of x Inverse Function: By definition Restrictions: None Odd/Even: Odd General Form:

() = (( - )) + Algebraically: Swap , then solve for Graphically: Rotate about 45 line =

Conic Sections

Parabola

= 2

Copyright ? 2011-2019 by Harold A. Toomey, WyzAnt Tutor

Domain: (-, ) Range: [, ) or (-, ] Inverse Function:

-1 () = Restrictions: None Odd/Even: Even Vertex : (, ) Focus : (, + ) General Forms:

( - )2 = 4( - )

2 + + 2 + + + = 0 where 2 - 4 = 0

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Function Parent Name Function

Circle

2 + 2 = 2

Ellipse

2 2 2 + 2 = 1

Hyperbola

2 2 2 - 2 = 1

Graph

Characteristics

Domain: [- + , + ] Range: [- + , + ] Inverse Function: Same as parent Restrictions: None Odd/Even: Both Focus : (, ) General Forms:

( - )2 + ( - )2 = 2

2 + + 2 + + + = 0 = = 0

Domain: [- + , + ] Range: [- + , + ] Inverse Function:

2 2 2 + 2 = 1 Restrictions: None Odd/Even: Both Foci : 2 = 2 - 2 General Forms: ( - )2 ( - )2 2 + 2 = 1

2 + + 2 + + + = 0 where 2 - 4 < 0

Domain: (-, -a+h] [a+h, ) Range: (-, ) Inverse Function:

2 2 2 - 2 = 1 Restrictions: Domain is restricted Odd/Even: Both Foci : 2 = 2 + 2 General Forms: ( - )2 ( - )2 2 - 2 = 1

2 + + 2 + + + = 0 where 2 - 4 > 0

Copyright ? 2011-2019 by Harold A. Toomey, WyzAnt Tutor

4

Function Name

Trigonometry

Parent Function

Sine

() =

Graph

Cosine

() =

Tangent

() =

=

Cosecant

() = 1

=

Secant

() = sec 1

=

Cotangent

() = 1

=

Copyright ? 2011-2019 by Harold A. Toomey, WyzAnt Tutor

Characteristics

Domain: (-, ) Range: [-1, 1] Inverse Function: -1 () = -1 Restrictions: None Odd/Even: Odd General Form:

() = (( - )) +

Domain: (-, ) Range: [-1, 1] Inverse Function: -1 () = -1 Restrictions: None Odd/Even: Even General Form:

() = (( - )) +

Domain:

(-,

)

except

for

=

2

?

Range: (-, )

Inverse Function: -1 () = -1

Restrictions:

Asymptotes

at

=

2

?

Odd/Even: Odd

General Form:

() = (( - )) +

Domain: (-, ) except for = ? Range: (-, -1] [1, ) Inverse Function: -1 () = -1 Restrictions: Range is bounded Odd/Even: Odd General Form:

() = (( - )) +

Domain:

(-,

)

except

for

=

2

?

Range: (-,-1] [1, )

Inverse Function: -1 () = -1

Restrictions: Range is bounded

Odd/Even: Even

General Form:

() = (( - )) +

Domain: (-, ) except for = ? Range: (-, ) Inverse Function: -1 () = -1 Restrictions: Asymptotes at x = ? Odd/Even: Odd General Form:

() = (( - )) +

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Function Parent Name Function

Arcsine

() = -1

Graph

Arccosine

() = -1

Arctangent

() = -1

Arccosecant

() = -1

Arcsecant

() = -1

Arccotangent () = -1 Copyright ? 2011-2019 by Harold A. Toomey, WyzAnt Tutor

Characteristics

Domain: [-1, 1]

Range:

[-

2

,

]

2

or

Quadrants

I

&

IV

Inverse Function: -1 () =

Restrictions: Range & Domain are bounded

Odd/Even: Odd

General Form:

() = -1 (( - )) +

Domain: [-1, 1] Range: [0, ] or Quadrants I & II Inverse Function: -1 () =

Restrictions: Range & Domain are bounded

Odd/Even: None

General Form: () = -1 (( - )) +

Domain: (-, )

Range: (-2 , 2) or Quadrants I & IV Inverse Function: -1 () =

Restrictions: Range is bounded

Odd/Even: Odd

General Form:

() = -1 (( - )) +

Domain: (-,-1] [1, )

Range:

[

-

2

,

0)

(0,

2]

or

Quadrants

I

&

IV

Inverse Function: -1 () =

Restrictions: Range & Domain are bounded

Odd/Even: Odd

General Form:

() = -1 (( - )) +

Domain: (-,-1] [1, )

Range:

[0,

)

2

(2

,

]

or

Quadrants

I

&

II

Inverse Function: -1 () =

Restrictions: Range & Domain are bounded

Odd/Even: Neither

General Form:

() = -1 (( - )) +

Domain: (-, ) Range: (0, ) or Quadrants I & II Inverse Function: -1 () =

Restrictions: Range is bounded Odd/Even: Neither

General Form: () = -1 (( - )) +

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Function Name

Hyperbolics

Parent Function

Hyperbolic Sine

() = sinh - -

= 2

Graph

Hyperbolic Cosine

() = + -

= 2

Hyperbolic Tangent

() = 2 - 1

= 2 + 1

Hyperbolic Cosecant

() = 1

=

Hyperbolic Secant

() = sech 1

=

Hyperbolic Cotangent

() = 1

=

Copyright ? 2011-2019 by Harold A. Toomey, WyzAnt Tutor

Characteristics

Domain: (-, ) Range: (-, ) Inverse Function: -1 () = -1 Restrictions: None Odd/Even: Odd General Form:

() = (( - )) +

Domain: (-, ) Range: [1, ) Inverse Function: -1 () = -1 Restrictions: None Odd/Even: Even General Form:

() = (( - )) +

Domain: (-, ) Range: (-1, 1) Inverse Function: -1 () = -1 Restrictions: Asymptotes at = ?1 Odd/Even: Odd General Form:

() = (( - )) +

Domain: (-, 0) (0, ) Range: (-, 0] [0, ) Inverse Function: -1 () = -1 Restrictions: Asymptotes at = 0, = 0 Odd/Even: Odd General Form:

() = (( - )) +

Domain: (-, ) Range: (0, 1] Inverse Function: -1 () = -1 Restrictions: Asymptote at = 0 Odd/Even: Even General Form:

() = (( - )) + Domain: (-, 0) (0, ) Range: (-, 1) (1, ) Inverse Function: -1 () = -1 Restrictions: Asymptotes at = 0, = ?1 Odd/Even: Odd General Form:

() = (( - )) +

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Function Parent Name Function

Hyperbolic Arcsine

() = -1 = ( + 2 + 1)

Graph

Hyperbolic Arccosine

() = -1 = ( + 2 - 1)

Hyperbolic Arctangent

() = -1 1 1 +

= 2 (1 - )

Hyperbolic Arccosecant

() = -1

=

1 (

+

12

+

1)

Hyperbolic Arcsecant

() = -1

=

1 (

+

1 2

-

1)

Hyperbolic Arccotangent

() = -1 1 + 1

= 2 ( - 1)

Copyright ? 2011-2019 by Harold A. Toomey, WyzAnt Tutor

Characteristics

Domain: (-, ) Range: (-, ) Inverse Function: -1 () = Restrictions: None Odd/Even: Odd General Form:

() = -1 (( - )) +

Domain: [1, ) Range: [0, ) Inverse Function: -1 () = Restrictions: 0 Odd/Even: Neither General Form:

() = -1 (( - )) +

Domain: (-1, 1) Range: (-, ) Inverse Function: -1 () = Restrictions: Asymptotes at = ?1 Odd/Even: Odd General Form:

() = -1 (( - )) +

Domain: (-, 0) (0, ) Range: (-, 0] [0, ) Inverse Function: -1 () = Restrictions: Asymptotes at = 0, = 0 Odd/Even: Odd General Form:

() = -1 (( - )) +

Domain: (0, 1] Range: [0, ) Inverse Function: -1 () = Restrictions: Odd/Even: Neither General Form:

() = -1 (( - )) +

Domain: [-, -1) (1, ] Range: (-, 0) (0, ) Inverse Function: -1 () = Restrictions: Asymptotes at = 0, = ?1 Odd/Even: Odd General Form:

() = -1 (( - )) +

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