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A formula sheet that lists most if not all Pre-Calculus Formulas.

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- 1. PreCalculus Formulas Sequences and Series: Binomial Theorem ⎛ n⎞ ( a + b) = ∑ ⎜ ⎟ a n− k b k k =0 ⎝ k ⎠ n n Find the rth term ⎛ n ⎞ n−( r −1) r −1 b ⎜ r − 1⎟ a ⎝ ⎠ Functions: To find the inverse function: 1. Set function = y 2. Interchange the variables 3. Solve for y Arithmetic Last Term Geometric Last Term an = a1 + (n − 1)d an = a1r Geometric Partial Sum Arithmetic Partial Sum ⎛1− rn ⎞ S n = a1 ⎜ ⎟ ⎝ 1− r ⎠ ⎛a +a ⎞ Sn = n ⎜ 1 n ⎟ ⎝ 2 ⎠ f -1 (x) n −1 Composition of functions: ( f g )( x) = f ( g ( x)) ( g f )( x) = g ( f ( x)) (f [r (cosθ + i sin θ )]n = r n (cos niθ + i sin niθ ) r = a 2 + b2 b θ = arctan a x = r cosθ y = r sin θ a + bi i = −1 i 2 = −1 ( r ,θ ) → ( x , y ) Determinants: 3 5 4 3 = 3i3 − 5i4 Use your calculator for 3x3 determinants. −1 f )( x) = x Algebra of functions: ( f + g )( x) = f ( x) + g ( x) ; ( f − g )( x) = f ( x) − g ( x) ( f i g )( x) = f ( x)i g ( x) ; ( f / g )( x) = f ( x) / g ( x), g ( x) ≠ 0 Domains:: D( f ( x)) ∩ D( g ( x)) Domain (usable x’s) Asymptotes: (vertical) Watch for problems with Check to see if the zero denominators and with denominator could ever be negatives under radicals. zero. x f ( x) = 2 Range (y’s used) x + x−6 Difference Quotient Vertical asymptotes at f ( x + h) − f ( x ) x = -3 and x = 2 h terms not containing a mult. of h will be eliminated. Complex and Polars: DeMoivre’s Theorem: Asymptotes: (horizontal) x+3 1. f ( x) = 2 x −2 top power < bottom power means y = 0 (z-axis) 4 x2 − 5 2. f ( x) = 2 3x + 4 x + 6 top power = bottom power means y = 4/3 (coefficients) x3 3. f ( x) = None! x+4 top power > bottom power All Rights Reserved © MathBits.com Cramer’s Rule: ax + by = c dx + ey = f ⎛c ⎜ a b⎝ f d e 1 b a , e d c⎞ ⎟ f ⎠ Also apply Cramer’s rule to 3 equations with 3 unknowns. Trig: Reference Triangles: o a o sin θ = ; cos θ = ; tan θ = h h a h h a csc θ = ; secθ = ; cotθ = o a o BowTie
- 2. Analytic Geometry: Circle Ellipse ( x − h) 2 + ( y − k ) 2 = r 2 Remember “completing the square” process for all conics. Parabola ( x − h) = 4a ( y − k ) ( y − k ) 2 = 4a ( x − h) 2 Polynomials: Remainder Theorem: Substitute into the expression to find the remainder. vertex to focus = a, length to directrix = a, latus rectum length from focus = 2a ( x − h) 2 ( y − k ) 2 + =1 a2 b2 larger denominator → major axis and smaller denominator → minor axis Hyperbola ( x − h) ( y − k) − =1 2 a b2 2 2 Latus length from focus b2/a c → focus length where major length is hypotenuse of right triangle. Latus rectum lengths from focus are b2/a a→transverse axis b→conjugate axis c→focus where c is the hypotenuse. asymptotes needed Descartes’ Rule of Signs 1. Maximum possible # of positive roots → number of sign changes in f (x) 2. Maximum possible # of negative roots → number of sign changes in f (-x) Depress equation [when dividing by (x - 5), use +5 for synthetic division] [(x + 3) substitutes -3] Synthetic Division Mantra: “Bring down, multiply and add, multiply and add…” (also use calculator to examine roots) Analysis of Roots P N C Chart Upper bounds: All values in chart are + Lower bounds: Values alternate signs No remainder: Root * all rows add to the degree * complex roots come in conjugate pairs * product of roots - sign of constant (same if degree even, opposite if degree odd) * decrease P or N entries by 2 −b ± b 2 − 4ac x= 2a Sum of roots is the coefficient of second term with sign changed. Eccentricity: e = 0 circle 0 < e < 1 ellipse e = 1 parabola e > 1 hyperbola Induction: Find P(1): Assume P(k) is true: Show P(k+1) is true: Rate of Growth/Decay: y = y0 ekt y = end result, y0 = start amount, Be sure to find the value of k first. Far-left/Far-right Behavior of a Polynomial The leading term (anxn ) of the polynomial determines the far-left/far-right behavior of the graph according to the following chart. (“Parity” of n whether n is odd or even.) LEFT-HAND BEHAVIOR n is even n is odd anxn (same as right) an > 0 RIGHTHAND always positive BEHAVIOR or (opposite right) negative x < 0 positive x > 0 Leading Coefficient Test Product of roots is the constant term (sign changed if odd degree, unchanged if even degree). All Rights Reserved © MathBits.com an < 0 always negative positive x < 0 negative x > 0

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