Information about Formula Sheet For Pre-Calculus

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

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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