Function Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015

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1 Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax + By + C 0 y mx + b y y 0 m(x x 0 ) Quadratic Square Square Root f(x) x f(x) x g(x) x Ax + By + Cx + D 0 Domain: [0, ) g(x) x Restrictions: x 0 f(x) a b(x h) + k Copyright 0-05 by Harold Toomey, WyzAnt Tut

2 Absolute Value f(x) x Cubic f(x) x 3 Cube Root Exponential Logarithmic 3 f(x) x f(x) 0 x f(x) e x f(x) log x f(x) ln x f(x) x f x 0 Restrictions: x, if x 0 f(x) { x, if x < 0 f(x) a b(x h) + k 3 g(x) x f(x) a(b(x h)) 3 + k g(x) x 3 3 f(x) a b(x h) + k Range: (0, ) g(x) log x g(x) ln x, x can be imaginary f(x) a 0 (b(x h)) + k Domain: (0, ) g(x) 0 x g(x) e x Restrictions: x > 0 f(x) a log(b(x h)) + k Copyright 0-05 by Harold A. Toomey, WyzAnt Tut

3 Domain: (, 0) (0, ) Range: (, 0) (0, ) Reciprocal Rational f(x) x g(x) x Restrictions: x 0 b f(x) a [ (x h) ] + k Greatest Integer Flo f(x) [x] whole numbers only Undefined (asymptotic) Restrictions: Real numbers only f(x) a[b(x h)] + k Inverse s If f(x) y, then f (y) f (f(x)) x Domain of x Domain of y Range of y Range of x By definition f(x) a f(b(x h)) + k Conic Sections Circle x + y r Domain: [ r + h, r + h] Range: [ r + k, r + k] Same as parent Odd/Even: Both Focus : (h, k) General Fms: (x h) + (y k) r Ax + Bxy + Cy + Dx + Ey + F 0 where A C and B 0 Copyright 0-05 by Harold A. Toomey, WyzAnt Tut 3

4 Ellipse x a + y b Domain: [ a + h, a + h] Range: [ b + k, b + k] x b + y a Odd/Even: Both Foci : c a b General Fms: (x h) a (y k) + b Ax + Bxy + Cy + Dx + Ey + F 0 where B 4AC < 0 Parabola y ax Range: [k, ) (, k] g(x) x Vertex : (h, k) Focus : (h, k + p) General Fms: (x h) 4p(y k) Hyperbola x a y b Ax + Bxy + Cy + Dx + Ey + F 0 where B 4AC 0 Domain: (, -a+h] [a+h, ) y a x b Restrictions: Domain is restricted Odd/Even: Both Foci : c a + b General Fms: (x h) a (y k) b Ax + Bxy + Cy + Dx + Ey + F 0 where B 4AC > 0 Copyright 0-05 by Harold A. Toomey, WyzAnt Tut 4

5 Trigonometry Sine Cosine f(x) sin x f(x) cos x Range: [, ] g(x) sin x f(x) a sin (b(x h)) + k Range: [, ] g(x) cos x f(x) a cos (b(x h)) + k Tangent f(x) tan x sin x cos x except f x π ± nπ g(x) tan x Restrictions: Asymptotes at x π ± nπ f(x) a tan (b(x h)) + k Secant f(x) sec x cos x except f x π ± nπ Range: (, ] [, ) g(x) sec x Restrictions: Range is bounded f(x) a sec (b(x h)) + k Cosecant f(x) csc x sin x except f x ±nπ Range: (, -] [, ) g(x) csc x Restrictions: Range is bounded f(x) a csc (b(x h)) + k Cotangent f(x) cot x tan x except f x ±nπ g(x) cot x Restrictions: Asymptotes at x ±nπ f(x) a cot (b(x h)) + k Copyright 0-05 by Harold A. Toomey, WyzAnt Tut 5

6 Arcsine Arccosine Arctangent Arcsecant Arccosecant f(x) sin x f(x) cos x f(x) tan x f(x) sec x f(x) csc x Domain: [, ] Range: [ π, π ] Quadrants I & IV g(x) sin x Restrictions: Range & Domain are bounded f(x) a sin (b(x h)) + k Domain: [, ] Range: [0, π] Quadrants I & II g(x) cos x Restrictions: Range & Domain are bounded Odd/Even: None f(x) a cos (b(x h)) + k Range: ( π, π ) Quadrants I & IV g(x) tan x Restrictions: Range is bounded f(x) a tan (b(x h)) + k Domain: (, ] [, ) Range: [0, π ) (π, π] Quadrants I & II g(x) sec x Restrictions: Range & Domain are bounded f(x) a sec (b(x h)) + k Domain: (, ] [, ) Range: [ π, 0) (0, π ] Quadrants I & IV g(x) csc x Restrictions: Range & Domain are bounded f(x) a csc (b(x h)) + k Arccotangent f(x) cot x Range: (0, π) Quadrants I & II g(x) cot x Restrictions: Range is bounded f(x) a cot (b(x h)) +k Copyright 0-05 by Harold A. Toomey, WyzAnt Tut 6

7 s Sine f(x) sinh x ex e x g(x) sinh x f(x) a sinh (b(x h)) + k Cosine f(x) cosh x ex + e x Range: [, ) g(x) cosh x f(x) a cosh (b(x h)) + k Tangent f(x) tanh x ex e x + Range: (, ) g(x) tanh x Restrictions: Asymptotes at y ± f(x) a tanh (b(x h)) + k Secant Cosecant f(x) sech x cosh x f(x) csch x sinh x Range: (0, ] g(x) sech x Restrictions: Asymptote at y 0 f(x) a sech (b(x h)) + k Domain: (, 0) (0, ) Range: (, 0] [0, ) g(x) csch x Restrictions: Asymptotes at x 0, y 0 f(x) a csch (b(x h)) + k Cotangent f(x) coth x ex + e x Domain: (, 0) (0, ) Range: (, ) (, ) g(x) coth x Restrictions: Asymptotes at x 0, y ± f(x) a coth (b(x h)) + k Copyright 0-05 by Harold A. Toomey, WyzAnt Tut 7

8 Arcsine f(x) sinh x ln(x + x + ) g(x) sinh x f(x) a sinh (b(x h)) + k Arccosine f(x) cosh x ln(x + x ) Domain: [, ) g(x) cosh x Restrictions: y 0 f(x) a cosh (b(x h)) + k Arctangent f(x) tanh x + x ln ( x ) Domain: (, ) g(x) tanh x Restrictions: Asymptotes at x ± f(x) a tanh (b(x h)) + k Arcsecant f(x) sech x ln ( x + x ) Domain: (0, ] g(x) sech x Restrictions: f(x) a sech (b(x h)) + k Arccosecant f(x) csch x ln ( x + x + ) Domain: (, 0) (0, ) Range: (, 0] [0, ) g(x) csch x Restrictions: Asymptotes at x 0, y 0 f(x) a csch (b(x h)) + k Arccotangent f(x) coth x + ln (x x ) Domain: [, ) (, ] Range: (, 0) (0, ) g(x) coth x Restrictions: Asymptotes at x 0, y ± f(x) a coth (b(x h)) +k Copyright 0-05 by Harold A. Toomey, WyzAnt Tut 8

9 ing Tips All s The Six Levers y a f (b (x - h)) + k ing Tips ) Move up/down k (Vertical translation) + Moves it up ) Move left/right h (Phase shift) + Moves it right 3) Stretch up/down a (Amplitude) Larger stretches it taller makes it grow faster 4) Stretch left/right b (Frequency π) Larger stretches it wider 5) Flip about x-axis a a 6) Flip about y-axis b b f(x) f(x) If f(x) f( x) then odd function f(x) f( x) If f(x) f( x) then even function 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) If k f(x) then x-axis is replaced by f(x)-axis ) Move left/right h (Phase shift) + shifts right sin (x) cos (x π/) 3) Stretch up/down a (Amplitude) (max min) a 4) Stretch left/right b (Frequency π) T π b ƒ a is NOT peak-to-peak on y-axis T peak-to-peak on θ-axis T π f tan (bx) b 5) Flip about x-axis a a f(x) f( x) Odd : sin (x) sin ( x) 6) Flip about y-axis b b f(x) f( x) Even : cos (x) cos ( x) Copyright 0-05 by Harold Toomey, WyzAnt Tut 9