Parent Function. Function Name Algebra. Characteristics. Harold s Parent Functions Cheat Sheet 20 September 2016

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1 Harold s s Cheat Sheet 20 September 206 Algebra Constant Linear or Identity Quadratic or Square Square Root Domain: (, ) Range: [c, c] Undefined (asymptote) Restrictions: c is a real number 0 Domain: (, ) Range: (, ) Restrictions: m 0 0 Domain: (, ) 0 Domain: [0, ) x Restrictions: 0 Copyright by Harold Toomey, WyzAnt Tutor

2 Absolute Value Cubic Cube Root Exponential Logarithmic 0 log ln Domain: (, ) 0 Restrictions:, 0, 0 Domain: (, ) Range: (, ) Domain: (, ) Range: (, ) Domain: (, ) Range: (0, ) log ln, x can be imaginary 0 Domain: (0, ) Range: (, ) 0 Restrictions: x > 0 log Copyright by Harold A. Toomey, WyzAnt Tutor 2

3 Domain: (, 0) (0, ) Range: (, 0) (0, ) Reciprocal or Rational Restrictions: x 0 Greatest Integer or Floor Domain: (, ) Range: (, ) whole numbers only Undefined (asymptotic) Restrictions: Real numbers only Inverse s, Domain of x Domain of y Range of y Range of x By definition Conic Sections Circle Domain:, Range:, Same as parent Odd/Even: Both Focus :, 0 0 Copyright by Harold A. Toomey, WyzAnt Tutor 3

4 Ellipse Domain:, Range:, Odd/Even: Both Foci : 0 where 4 0 Parabola Domain: (, ) Range:, or, Vertex :, Focus :, 4 0 where 4 0 Hyperbola Domain: (, a+h] [a+h, ) Range: (, ) Restrictions: Domain is restricted Odd/Even: Both Foci : 0 where 4 0 Copyright by Harold A. Toomey, WyzAnt Tutor 4

5 Trigonometry Sine Cosine Domain: (, ) Range: [, ] Domain: (, ) Range: [, ] Tangent Domain: (, ) except for Range: (, ) Restrictions: Asymptotes at Secant sec Domain: (, ) except for Range: (,] [, ) Restrictions: Range is bounded Cosecant Domain: (, ) except for Range: (, ] [, ) Restrictions: Range is bounded Cotangent Domain: (, ) except for Range: (, ) Restrictions: Asymptotes at x = Copyright by Harold A. Toomey, WyzAnt Tutor 5

6 Arcsine Arccosine Arctangent Arcsecant Arccosecant Domain: [, ] Range:, or Quadrants I & IV Restrictions: Range & Domain are bounded Domain: [, ] Range: 0, or Quadrants I & II Restrictions: Range & Domain are bounded Odd/Even: None Domain: (, ) Range:, or Quadrants I & IV Restrictions: Range is bounded Domain: (,] [, ) Range: 0, (, or Quadrants I & II Restrictions: Range & Domain are bounded Domain: (,] [, ) Range:,0 0, or Quadrants I & IV Restrictions: Range & Domain are bounded Arccotangent Domain: (, ) Range: 0, or Quadrants I & II Restrictions: Range is bounded Copyright by Harold A. Toomey, WyzAnt Tutor 6

7 s Sine sinh 2 Domain: (, ) Range: (, ) Cosine 2 Domain: (, ) Range: [, ) Tangent Domain: (, ) Range: (, ) Restrictions: Asymptotes at Secant Cosecant sech Domain: (, ) Range: (0, ] Restrictions: Asymptote at 0 Domain: (, 0) (0, ) Range: (, 0] [0, ) Restrictions: Asymptotes at 0,0 Cotangent Domain: (, 0) (0, ) Range: (, ) (, ) Restrictions: Asymptotes at 0, Copyright by Harold A. Toomey, WyzAnt Tutor 7

8 Arcsine Domain: (, ) Range: (, ) Arccosine Domain: [, ) Restrictions: 0 Arctangent 2 Domain: (, ) Range: (, ) Restrictions: Asymptotes at Arcsecant Domain: (0, ] Restrictions: Arccosecant Domain: (, 0) (0, ) Range: (, 0] [0, ) Restrictions: Asymptotes at 0,0 Arccotangent 2 Domain:,, Range:, 0 0, Restrictions: Asymptotes at 0, Copyright by Harold A. Toomey, WyzAnt Tutor 8

9 ing Tips All s The Seven Levers y = a f (b (x h)) + k ing Tips ) 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 wider 5) Flip about x axis a a 6) Flip about y axis b b 7) Rotate CW/CCW cot 2θ AC B If then odd function If then even function + θ rotates CCW For conic sections, where: 0 Trigonometric s The Six Trig Levers y = a sin (b (x h)) + k ing Tips Notes ) 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) max min a 2 4) Stretch left/right b (Frequency 2π) T 2π b ƒ a is NOT peak to peak on y axis T = peak to peak on θ axis for 5) Flip about x axis a a Odd : 6) Flip about y axis b b Even : Copyright by Harold Toomey, WyzAnt Tutor 9