Function Parent Graph Characteristics Name Function

1 Copyright 2011-2019 by Harold Toomey, WyzAnt Tutor 1 Harold s Parent functions Cheat Sheet 6 November 2019 Function Name Parent Function Graph Characteristics Algebra Constant ( )= Domain: ( , ) Range: [c, c] Inverse Function : Undefined (asymptote) Restrictions: c is a real number Odd/Even: Even General Form: + =0 Linear or Identity ( )= Domain: ( , ) Range: ( , ) Inverse Function : Same as Parent Restrictions: m 0 Odd/Even: Odd General Forms: + + =0 = + 0= ( 0) Quadratic or Square ( )= 2 Domain: ( , ) Range: [0, ) Inverse Function : 1 ( )= Restrictions: None Odd/Even: Even General Form: 2+ + + =0 Square Root ( )= Domain: [0, ) Range: [0, ) Inverse Function : 1 ( )= x2 Restrictions: 0 Odd/Even: Neither General Form: ( )= ( )+ Function Name Parent Function Graph Characteristics Copyright 2011-2019 by Harold A.]]]

2 Toomey, WyzAnt Tutor 2 Cubic ( )= 3 Domain: ( , ) Range: ( , ) Inverse Function : 1 ( )= 3 Restrictions: None Odd/Even: Odd General Form: ( )= ( ( ))3+ Cube Root ( )= 3 Domain: ( , ) Range: ( , ) Inverse Function : 1 ( )= 3 Restrictions: None Odd/Even: Odd General Form: ( )= ( )3+ Reciprocal or Rational ( )= 1 Domain: ( , 0) (0, ) Range: ( , 0) (0, ) Inverse Function : Same as Parent Restrictions: x 0 Odd/Even: Odd General Form: ( )= ( )+ Transcendentals Exponential ( )=10 ( )= Domain: ( , ) Range: (0, ) Inverse Function : 1 ( )=log 1 ( )=ln Restrictions: None, x can be complex Odd/Even: Neither General Form: ( )= 10( ( ))+ Logarithmic ( )=log ( )=ln Domain: (0, ) Range: ( , ) Inverse Function : 1 ( )=10 1 ( )= Restrictions: x > 0 Odd/Even: Neither General Form: ( )= log( ( ))+ Function Name Parent Function Graph Characteristics Copyright 2011-2019 by Harold A.

3 Toomey, WyzAnt Tutor 3 Absolute Value ( )=| | Domain: ( , ) Range: [0, ) Inverse Function : 1 ( )= 0 Restrictions: ( )={ , 0 , <0 Odd/Even: Even 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 ( 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 Domain: ( , ) Range: [ , ) or ( , ] Inverse Function : 1 ( )= Restrictions: None Odd/Even: Even Vertex : ( , ) Focus : ( , + ) General Forms: ( )2=4 ( ) 2+ + 2+ + + =0 where 2 4 =0 Function Name Parent Function Graph Characteristics Copyright 2011-2019 by Harold A.]}

4 Toomey, WyzAnt Tutor 4 Circle 2+ 2= 2 Domain: [ + , + ] Range: [ + , + ] Inverse Function : Same as Parent Restrictions: None Odd/Even: Both Focus : ( , ) General Forms: ( )2+( )2= 2 2+ + 2+ + + =0 = =0 Ellipse 2 2+ 2 2=1 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 Hyperbola 2 2 2 2=1 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 Function Name Parent Function Graph Characteristics Copyright 2011-2019 by Harold A.

5 Toomey, WyzAnt Tutor 5 Trigonometry Sine ( )= Domain: ( , ) Range: [ 1, 1] Inverse Function : 1 ( )= 1 Restrictions: None Odd/Even: Odd General Form: ( )= ( ( ))+ Cosine ( )= Domain: ( , ) Range: [ 1, 1] Inverse Function : 1 ( )= 1 Restrictions: None Odd/Even: Even General Form: ( )= ( ( ))+ Tangent ( )= = Domain: ( , ) except for = 2 Range: ( , ) Inverse Function : 1 ( )= 1 Restrictions: Asymptotes at = 2 Odd/Even: Odd General Form: ( )= ( ( ))+ Cosecant ( )= = 1 Domain: ( , ) except for = Range: ( , -1] [1, ) Inverse Function : 1 ( )= 1 Restrictions: Range is bounded Odd/Even: Odd General Form: ( )= ( ( ))+ Secant ( )=sec = 1 Domain: ( , ) except for = 2 Range: ( , 1] [1, ) Inverse Function : 1 ( )= 1 Restrictions: Range is bounded Odd/Even: Even General Form: ( )= ( ( ))+ Cotangent ( )= = 1 Domain: ( , ) except for = Range: ( , ) Inverse Function : 1 ( )= 1 Restrictions: Asymptotes at x = Odd/Even: Odd General Form: ( )= ( ( ))+ Function Name Parent Function Graph Characteristics Copyright 2011-2019 by Harold A.

6 Toomey, WyzAnt Tutor 6 Arcsine ( )= 1 Domain: [ 1, 1] Range: [ 2, 2] or Quadrants I & IV Inverse Function : 1 ( )= Restrictions: Range & Domain are bounded Odd/Even: Odd General Form: ( )= 1 ( ( ))+ Arccosine ( )= 1 Domain: [ 1, 1] Range: [0, ] or Quadrants I & II Inverse Function : 1 ( )= Restrictions: Range & Domain are bounded Odd/Even: None General Form: ( )= 1 ( ( ))+ Arctangent ( )= 1 Domain: ( , ) Range: ( 2, 2) or Quadrants I & IV Inverse Function : 1 ( )= Restrictions: Range is bounded Odd/Even: Odd General Form: ( )= 1 ( ( ))+ Arccosecant ( )= 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 ( ( ))+ Arcsecant ( )= 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 ( ( ))+ Arccotangent ( )= 1 Domain: ( , ) Range: (0, ) or Quadrants I & II Inverse Function : 1 ( )= Restrictions: Range is bounded Odd/Even: Neither General Form: ( )= 1 ( ( ))+ Function Name Parent Function Graph Characteristics Copyright 2011-2019 by Harold A.

7 Toomey, WyzAnt Tutor 7 Hyperbolics Hyperbolic Sine ( )=sinh = 2 Domain: ( , ) Range: ( , ) Inverse Function : 1 ( )= 1 Restrictions: None Odd/Even: Odd General Form: ( )= ( ( ))+ Hyperbolic Cosine ( )= = + 2 Domain: ( , ) Range: [1, ) Inverse Function : 1 ( )= 1 Restrictions: None Odd/Even: Even General Form: ( )= ( ( ))+ Hyperbolic Tangent ( )= = 2 1 2 +1 Domain: ( , ) Range: ( 1, 1) Inverse Function : 1 ( )= 1 Restrictions: Asymptotes at = 1 Odd/Even: Odd General Form: ( )= ( ( ))+ Hyperbolic Cosecant ( )= = 1 Domain: ( , 0) (0, ) Range: ( , 0] [0, ) Inverse Function : 1 ( )= 1 Restrictions: Asymptotes at =0, =0 Odd/Even: Odd General Form: ( )= ( ( ))+ Hyperbolic Secant ( )=sech = 1 Domain: ( , ) Range: (0, 1] Inverse Function : 1 ( )= 1 Restrictions: Asymptote at =0 Odd/Even: Even General Form: ( )= ( ( ))+ Hyperbolic Cotangent ( )= = 1 Domain: ( , 0) (0, ) Range: ( , 1) (1, ) Inverse Function : 1 ( )= 1 Restrictions: Asymptotes at =0, = 1 Odd/Even: Odd General Form: ( )= ( ( ))+ Function Name Parent Function Graph Characteristics Copyright 2011-2019 by Harold A.

8 Toomey, WyzAnt Tutor 8 Hyperbolic Arcsine ( )= 1 = ( + 2+1) Domain: ( , ) Range: ( , ) Inverse Function : 1 ( )= Restrictions: None Odd/Even: Odd General Form: ( )= 1 ( ( ))+ Hyperbolic Arccosine ( )= 1 = ( + 2 1) Domain: [1, ) Range: [0, ) Inverse Function : 1 ( )= Restrictions: 0 Odd/Even: Neither General Form: ( )= 1 ( ( ))+ Hyperbolic Arctangent ( )= 1 =12 (1+ 1 ) Domain: ( 1, 1) Range: ( , ) Inverse Function : 1 ( )= Restrictions: Asymptotes at = 1 Odd/Even: Odd General Form: ( )= 1 ( ( ))+ Hyperbolic Arccosecant ( )= 1 = (1 + 1 2+1) Domain: ( , 0) (0, ) Range: ( , 0] [0, ) Inverse Function : 1 ( )= Restrictions: Asymptotes at =0, =0 Odd/Even: Odd General Form: ( )= 1 ( ( ))+ Hyperbolic Arcsecant ( )= 1 = (1 + 1 2 1) Domain: (0, 1] Range: [0, ) Inverse Function : 1 ( )= Restrictions: Odd/Even: Neither General Form: ( )= 1 ( ( ))+ Hyperbolic Arccotangent ( )= 1 =12 ( +1 1) Domain: [ , 1) (1, ] Range: ( ,0) (0, ) Inverse Function : 1 ( )= Restrictions: Asymptotes at =0, = 1 Odd/Even: Odd General Form.]]

9 ( )= 1 ( ( ))+ Copyright 2011-2019 by Harold Toomey, WyzAnt Tutor 9 Graphing Tips All functions The Seven Function Levers y = a f (b (x - h)) + k Graphing Tips 1) Move up/down k (Vertical translation) + Moves it up 2) Move left/right h (Horizontal translation) + Moves it right 3) Stretch up/down a (Vertical dilation) Larger stretches it taller or makes it grow faster 4) Stretch left/right b (Horizontal dilation) Larger stretches it out wider 5) Flip about x-axis a a ( ) ( ) If ( )= ( ) then odd Function 6) Flip about y-axis b b ( ) ( ) If ( )= ( ) then even Function 7) Rotate CW/CCW cot2 =A CB + rotates CCW For conic sections, where: 2+ + 2+ + + =0 Trigonometric functions The Six Trig Levers y = a sin (b (x - h)) + k Graphing Tips Notes 1) Move up/down k (Vertical translation) k= (max + min)2 If = ( ) then x-axis is replaced by ( )-axis 2) Move left/right h (Phase shift) + shifts right ( )= ( /2) 3) Stretch up/down a (Amplitude) a= (max min)2 a is NOT peak-to-peak on y-axis 4) Stretch left/right b (Frequency 2 ) T=2 |b|= 1 T = peak-to-peak on -axis = for ( ) 5) Flip about y-axis b b ( ) ( ) Even Function : ( )= ( ) 6) Flip about x-axis a a ( ) ( ) Odd Function : ( )= ( )