Harolds Parent Functions Cheat Sheet 2016 PDF

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Summary

NOTES ON ELECTRICAL CIRCUIT ANALYSIS TEST2 RL RC AND RLC MESH ANALYSIS NODAL ANALYSIS KCL AND KVL SUPERPOSITIOON THEORM...

Description

Harold’s ParentFunctions “CheatSheet” 20September2016

 

Function Name Algebra

Parent Function 

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: 󰇛󰇜   Restrictions:m≠0 Odd/Even:Odd GeneralForms:       0           󰇛   󰇜



Constant

󰇛󰇜  

Linear or Identity

󰇛󰇜  

󰇛󰇜    

Quadratic or Square

SquareRoot

Graph

         

󰇛󰇜   √



Domain:(∞,∞) Range:[0,∞) InverseFunction: 󰇛󰇜   √ Restrictions:None Odd/Even:Even GeneralForm:          0 Domain:[0,∞) Range:[0,∞) InverseFunction: 󰇛󰇜   x   Restrictions:  0 Odd/Even:Neither GeneralForm: 󰇛󰇜  󰇛  󰇜  

Copyright©2011‐2016byHaroldToomey,WyzAntTutor1 



Function Name

Parent Function

Graph

Characteristics



AbsoluteValue

󰇛󰇜  ||

Cubic

󰇛󰇜    

CubeRoot

󰇛󰇜  √

Exponential

󰇛󰇜  10   󰇛󰇜    

Logarithmic

󰇛󰇜  log   󰇛󰇜  ln 

Domain:(∞,∞) Range:[0,∞) InverseFunction: 󰇛󰇜    0 Restrictions: ,  0  󰇛󰇜   ,  0 Odd/Even:Even  GeneralForm: 󰇛󰇜  |󰇛  󰇜|   Domain:(∞,∞) Range:(∞,∞) InverseFunction:  󰇛󰇜  √  Restrictions:None Odd/Even:Odd GeneralForm:  󰇛󰇜  󰇛󰇛  󰇜󰇜   Domain:(∞,∞) Range:(∞,∞) InverseFunction: 󰇛󰇜    Restrictions:None Odd/Even:Odd GeneralForm:    󰇛󰇜   󰇛  󰇜   Domain:(∞,∞) Range:(0,∞) InverseFunction: 󰇛󰇜  log   󰇛󰇜  ln  Restrictions:None,xcanbeimaginary Odd/Even:Neither GeneralForm:  󰇛󰇜  10󰇛󰇛󰇜󰇜   Domain:(0,∞) Range:(∞,∞) InverseFunction: 󰇛󰇜  10   󰇛󰇜     Restrictions:x>0 Odd/Even:Neither  GeneralForm: 󰇛󰇜   log󰇛󰇛  󰇜󰇜  

Copyright©2011‐2016byHaroldA.Toomey,WyzAntTutor2 



Function Name

Parent Function

Graph

Characteristics



1 󰇛󰇜    

Reciprocal or Rational



Greatest Integer or Floor

󰇛󰇜  󰇟󰇠

Inverse Functions

󰇛󰇜  ,   󰇛󰇜   󰇛󰇛󰇜󰇜  

 

ConicSections

Circle





     

      

Domain:(∞,0)∪(0,∞) Range:(∞,0)∪(0,∞) InverseFunction: 1 󰇛󰇜     Restrictions:x≠0 Odd/Even:Odd GeneralForm:   󰇛󰇜       󰇛  󰇜  Domain:(∞,∞) Range:(∞,∞)wholenumbersonly InverseFunction:Undefined(asymptotic) Restrictions:Realnumbersonly Odd/Even:Neither GeneralForm: 󰇛󰇜  󰇟󰇛  󰇜󰇠   DomainofxDomainofy RangeofyRangeofx InverseFunction:Bydefinition Restrictions:None Odd/Even:Odd GeneralForm: 󰇛󰇜  󰇛󰇛  󰇜󰇜  

 Domain:󰇟  ,   󰇠 Range:󰇟  ,   󰇠 InverseFunction:Sameasparent Restrictions:None Odd/Even:Both Focus:󰇛, 󰇜 GeneralForms: 󰇛  󰇜  󰇛  󰇜                   0      0 

Copyright©2011‐2016byHaroldA.Toomey,WyzAntTutor3 



Function Name

Parent Function

Graph

Characteristics





Ellipse 

   1    

Parabola

    



Hyperbola

   1   



Domain:󰇟  ,   󰇠 Range:󰇟  ,   󰇠 InverseFunction:    1    Restrictions:None Odd/Even:Both Foci:        GeneralForms: 󰇛  󰇜 󰇛  󰇜   1                 0 where   4  0  Domain:(∞,∞) Range:󰇟, ∞󰇜or󰇛∞, 󰇠 InverseFunction: 󰇛󰇜   √ Restrictions:None Odd/Even:Even Vertex:󰇛, 󰇜 Focus:󰇛,   󰇜 GeneralForms: 󰇛  󰇜  4󰇛  󰇜               0 where   4  0  Domain:(∞,‐a+h]∪[a+h,∞) Range:(∞,∞) InverseFunction:    1    Restrictions:Domainisrestricted Odd/Even:Both Foci:        GeneralForms: 󰇛  󰇜 󰇛  󰇜   1                  0 where   4  0 

   Copyright©2011‐2016byHaroldA.Toomey,WyzAntTutor4 



Function Name

Parent Function

Graph

Characteristics



Trigonometry





󰇛󰇜  

Sine



󰇛󰇜   

Cosine



Tangent



󰇛󰇜    

    



Secant





󰇛󰇜  sec  

Cosecant

1   

󰇛󰇜    1    



Cotangent



󰇛󰇜    

1   

Domain:(∞,∞)exceptfor     Range:(∞,∞) InverseFunction:󰇛󰇜     Restrictions:Asymptotesat     Odd/Even:Odd GeneralForm: 󰇛󰇜     󰇛  󰇜   

Domain:(∞,∞)exceptfor 

 

 

Range:(∞,1]∪[1,∞) InverseFunction:󰇛󰇜     Restrictions:Rangeisbounded Odd/Even:Even GeneralForm: 󰇛󰇜  󰇛  󰇜  





 Domain:(∞,∞) Range:[1,1] InverseFunction:󰇛󰇜    Restrictions:None Odd/Even:Odd GeneralForm: 󰇛󰇜  󰇛  󰇜   Domain:(∞,∞) Range:[1,1] InverseFunction:󰇛󰇜     Restrictions:None Odd/Even:Even GeneralForm: 󰇛󰇜    󰇛  󰇜  



Domain:(∞,∞)exceptfor   Range:(∞,‐1]∪[1,∞) InverseFunction:󰇛󰇜    Restrictions:Rangeisbounded Odd/Even:Odd GeneralForm: 󰇛󰇜    󰇛  󰇜  

Domain:(∞,∞)exceptfor   Range:(∞,∞) InverseFunction:󰇛󰇜     Restrictions:Asymptotesatx= Odd/Even:Odd GeneralForm: 󰇛󰇜    󰇛  󰇜  

Copyright©2011‐2016byHaroldA.Toomey,WyzAntTutor5 



Function Name

Parent Function

Graph

Characteristics



Domain:[1,1]

Arcsine

 

Range:󰇣  ,  󰇤orQuadrantsI&IV InverseFunction:󰇛󰇜    Restrictions:Range&Domainarebounded Odd/Even:Odd GeneralForm:  󰇛󰇜   󰇛  󰇜  

󰇛󰇜   



Arccosine

Domain:[1,1] Range:󰇟0, 󰇠orQuadrantsI&II InverseFunction:󰇛󰇜    Restrictions:Range&Domainarebounded Odd/Even:None GeneralForm: 󰇛󰇜    󰇛  󰇜   Domain:(∞,∞)

󰇛󰇜    

Arctangent

󰇛󰇜   

Arcsecant

󰇛󰇜   

Arccosecant

󰇛󰇜   

 

 



Arccotangent

Range:󰇡  ,  󰇢orQuadrantsI&IV InverseFunction:󰇛󰇜    Restrictions:Rangeisbounded Odd/Even:Odd GeneralForm: 󰇛󰇜    󰇛  󰇜   Domain:(∞,1]∪[1,∞)   Range:󰇟0,  󰇜∪(  , 󰇠orQuadrantsI&II InverseFunction:󰇛󰇜    Restrictions:Range&Domainarebounded Odd/Even:Neither GeneralForm:  󰇛󰇜   󰇛  󰇜   Domain:(∞,1]∪[1,∞)   Range:󰇟   , 0󰇜∪󰇛0,  󰇠orQuadrantsI&IV InverseFunction:󰇛󰇜    Restrictions:Range&Domainarebounded Odd/Even:Odd GeneralForm: 󰇛󰇜    󰇛  󰇜  

󰇛󰇜    



Domain:(∞,∞) Range:󰇛0, 󰇜orQuadrantsI&II InverseFunction:󰇛󰇜    Restrictions:Rangeisbounded Odd/Even:Neither GeneralForm:  󰇛󰇜    󰇛  󰇜 

   Copyright©2011‐2016byHaroldA.Toomey,WyzAntTutor6 



Function Name

Parent Function

Graph

Characteristics



Hyperbolics

HyperbolicSine







 Domain:(∞,∞) Range:(∞,∞) InverseFunction:󰇛󰇜    Restrictions:None Odd/Even:Odd GeneralForm: 󰇛󰇜  󰇛  󰇜  

󰇛󰇜  sinh  

      2



Hyperbolic Cosine



Domain:(∞,∞) Range:[1,∞) InverseFunction:󰇛󰇜    Restrictions:None Odd/Even:Even GeneralForm: 󰇛󰇜    󰇛  󰇜  

󰇛󰇜    

      2



Hyperbolic Tangent



Domain:(∞,∞) Range:(1,1) InverseFunction:󰇛󰇜    Restrictions:Asymptotesat  1 Odd/Even:Odd GeneralForm: 󰇛󰇜    󰇛  󰇜  

󰇛󰇜    

   1     1



Hyperbolic Secant



 󰇛󰇜  sech 

Hyperbolic Cosecant



󰇛󰇜   





1   

1

 

  



Hyperbolic Cotangent



󰇛󰇜    

   1     1



Domain:(∞,∞) Range:(0,1] InverseFunction:󰇛󰇜    Restrictions:Asymptoteat  0 Odd/Even:Even GeneralForm: 󰇛󰇜  󰇛  󰇜   Domain:(∞,0)∪(0,∞) Range:(∞,0]∪[0,∞) InverseFunction:󰇛󰇜    Restrictions:Asymptotesat  0,   0 Odd/Even:Odd GeneralForm: 󰇛󰇜   󰇛  󰇜  

Domain:(∞,0)∪(0,∞) Range:(∞,1)∪(1,∞) InverseFunction:󰇛󰇜    Restrictions:Asymptotesat  0,   1 Odd/Even:Odd GeneralForm: 󰇛󰇜   󰇛  󰇜  

Copyright©2011‐2016byHaroldA.Toomey,WyzAntTutor7 



Function Name

Parent Function

Graph

Characteristics



Hyperbolic



Arcsine

󰇛󰇜   

 󰇛     1󰇜 

Hyperbolic Arccosine



󰇛󰇜   

 󰇛     1󰇜 

Hyperbolic Arcsecant

Hyperbolic Arccosecant

 󰇛󰇜    

Domain:(0,1] Range:[0,∞) InverseFunction:󰇛󰇜    Restrictions: Odd/Even:Neither GeneralForm:  󰇛󰇜   󰇛  󰇜  

1 1   󰇭     1󰇮  



 󰇛󰇜   

1 1   󰇭     1󰇮   

Hyperbolic Arccotangent

󰇛󰇜     1 1     1 2

Domain:[1,∞) Range:[0,∞) InverseFunction:󰇛󰇜    Restrictions:  0 Odd/Even:Neither GeneralForm: 󰇛󰇜    󰇛  󰇜   Domain:(1,1) Range:(∞,∞) InverseFunction:󰇛󰇜    Restrictions:Asymptotesat  1 Odd/Even:Odd GeneralForm: 󰇛󰇜    󰇛  󰇜  

󰇛󰇜     1 1     2 1

Hyperbolic Arctangent



Domain:(∞,∞) Range:(∞,∞) InverseFunction:󰇛󰇜    Restrictions:None Odd/Even:Odd GeneralForm: 󰇛󰇜   󰇛  󰇜  

Domain:(∞,0)∪(0,∞) Range:(∞,0]∪[0,∞) InverseFunction:󰇛󰇜    Restrictions:Asymptotesat  0,   0 Odd/Even:Odd GeneralForm: 󰇛󰇜    󰇛  󰇜  

Domain:󰇟∞, 1󰇜∪󰇛1, ∞󰇠 Range:󰇛∞, 0󰇜∪󰇛0, ∞󰇜 InverseFunction:󰇛󰇜    Restrictions:Asymptotesat  0,   1 Odd/Even:Odd GeneralForm: 󰇛󰇜    󰇛  󰇜 

Copyright©2011‐2016byHaroldA.Toomey,WyzAntTutor8 

GraphingTips 

AllFunctions  TheSevenFunction “Levers”

y=af(b(x‐h))+k

GraphingTips

1) Moveup/down↕

k(Verticaltranslation)

2) Moveleft/right↔

h(Horizontaltranslation) “+“Movesitright

3) Stretchup/down↕

a(Verticaldilation)

4) Stretchleft/right↔ b(Horizontaldilation) 5) Flipaboutx‐axis

a→–a

6) Flipabouty‐axis

b→–b

7) RotateCW/CCW

cot 2θ 

  

AC  B

“+”Movesitup

Largerstretchesittallerormakesitgrowfaster Largerstretchesitwider

󰇛󰇜→– 󰇛󰇜 If󰇛󰇜 – 󰇛󰇜thenoddfunction 󰇛󰇜 →󰇛󰇜 If󰇛󰇜  󰇛󰇜thenevenfunction

“+”θrotatesCCW Forconicsections,where:               0

TrigonometricFunctions  TheSixTrig“Levers”

y=asin(b(x‐h))+k

1) Moveup/down↕

k(Verticaltranslation)

2) Moveleft/right↔

h(Phaseshift)

3) Stretchup/down↕

a(Amplitude)

4) Stretchleft/right↔ b(Frequency⦁2π) 5) Flipaboutx‐axis

a→–a

6) Flipabouty‐axis

b→–b

 

GraphingTips k

󰇛max  min󰇜  2

‘+‘shiftsright

󰇛max – min󰇜  2 1 2π   T |b| ƒ

a

󰇛󰇜→󰇛󰇜 󰇛󰇜→󰇛󰇜

Notes

If  󰇛󰇜thenx‐axisisreplaced by󰇛󰇜‐axis  󰇛󰇜  󰇛  /2󰇜

aisNOTpeak‐to‐peakony‐axis T=peak‐to‐peakonθ‐axis    for󰇛󰇜 

OddFunction: 󰇛󰇜   󰇛󰇜 EvenFunction: 󰇛󰇜   󰇛󰇜

Copyright©2011‐2016byHaroldToomey,WyzAntTutor9 ...