FormulaSheetAlgebra - Algebra – Things to Remember...

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Unformatted text preview: Algebra – Things to Remember! Scientific Notation: 3.2 x 1013 The first number must be 1 < n < 10 Factorial: 5! = 5•4•3•2•1 1! = 1 FYI: 0!=1 Absolute Value: | -5 | = 5 |5|=5 4−3 = Remember to −3 x ≤ 8 + x change direction −4 x ≤ 8 of inequality when x ≥ −2 mult/div by a negative. x = abscissa, y = ordinate Slope: vertical change rise y2 − y1 = = . m= horizontal change run x2 − x1 m+ n x •x = x ( x n ) m = x n• m m n xm = x m−n n x ( xy ) n = x n • y n 1 43 Represents distance Undefined: 6 is undefined when x = 7 since 7−x the denominator = 0. Multiply: (distribute or FOIL) ( x + 3)( x + 2) = x • x + x • 2 + 3• x + 3• 2 = x2 + 5x + 6 (a + b) 2 = a 2 + 2ab + b 2 (a − b) 2 = a 2 − 2ab + b 2 Add Fractions: Get the common denominator: 5 x 3x 5 x 9 x 14 x 7 x + = + = = 6 2 6 6 6 3 Inequalities: 5 − 3x ≤ 13 + x Exponents: (−3) 2 ≠ −32 20 = 1 Properties of Real Numbers: Commutative Property: a + b = b + a Associative Property: a+(b+c) = (a+b)+c Distributive Property: a(b+c) = ab + ac Identity: a+0=a Inverse: a + (-a) = 0 Zero Property: Polygons and sides: triangle – 3 octagon – 8 quadrilateral – 4 nonagon – 9 pentagon – 5 decagon – 10 hexagon – 6 dodecagon - 12 septagon – 7 ab = ba a(bc) = (ab)c a•1=a a • (1/a) = 1 a•0=0 Degree: Degree of monomial = sum of exponents 4x3 is of degree 3 x2y3 is of degree 5 Solving Equations: 1. Deal with any parentheses in the problem. 2. Combine similar terms on same side of = sign. Direct Variation: y = kx where k = constant of variation 3. Get the needed variables on the same side of = sign. 4. Isolate the needed variable by add or subtract. k = y/x 5. Find the needed variable by divide or multiply. Quadratic Equation: Interval Notation: Factor: 2 (1,5) ↔ 1 < x < 5 Look for a GCF (greatest common factor) x − 5x + 6 = 0 Set = 0. Factor binomial or trinomial. [1,5] ↔ 1 ≤ x ≤ 5 ( x − 3)( x − 2) = 0 Factor. a 2 − b 2 = (a + b)(a − b) x = 3; x = 2 Find roots Function: Passes the vertical line test. Parabola: Systems: A set of ordered pairs in which each x y = ax 2 + bx + c y – 2x = 1 Linear: substitute; element has only one y element y + 2x = 9 add to eliminate one Axis of symmetry: associated with it. variable or graph. −b x= f (x) = 3x + 4 y = x2 –x-6 Linear Quadratic: 2a f (3) = 3•3 + 4 = 13 y = 2x – 2 substitute or graph Roots: where the For inequality systems, graph. Parallel and Perpendicular: graph crosses the Parallel: slopes are equal. Equations of Lines: m = slope x-axis. Perpendicular: slopes are negative y = mx + b slope-intercept reciprocals (flip over and negate) y − y = m( x − x ) point-slope 1 1 Copyright © Regents Exam Prep Center http://regentsprep.org Perimeter: add the distances around the outside. Trig: Right triangles only Pythagorean Theorem: 2 2 2 o a o Right Triangles only. c = a + b sin A = ; cos A = ; tan A = Triples: 3, 4, 5 h h a Circumference: C = 2π r = π d 5, 12, 13 Angle of elevation: from horizontal line of sight up. 8, 15, 17 Angle of depression: from horizontal line of sight down. 7, 24, 25 Area: Volume and Surface Area: Data: 5 Statistical Summary: minimum, maximum, median, 1st quartile, 1 Vrectangular solid = l i wih Atriangle = bh 3rd quartile 2 SArectangular solid = 2lh + 2hw + 2lw Quartiles divide data into 4 equal parts. s2 3 Vcylinder = π r 2 h Percentiles divide data into 100 equal parts. Aequilateral triangle = 4 2 number of scores below x SAclosed cylinder = 2π rh + 2π r Percentile rank of score x = i100 , where n is Arectangle = bh n Error in Measurement: Asquare = bh = s 2 the number of scores. Relative error = |measure-actual| Mean = average. Aparallelogram = bh actual Mode = most often (may be more than one answer). d1 id 2 % of Error = Relative • 100% Median = middle. Arhombus = bh = 2 Outliers = values that are far away from the rest of the data. Permutations: 1 Median best describes data if outliers exist. Arrangement in specific order. Atrapezoid = h(b1 + b2 ) Range = difference between the maximum and minimum values. 2 n! 2 n Pr = Acircle = π r (n − r )! n Probability: P(A’) = 1 – P(A) complement Box and Whisker Plot: 1st and 3rd quartiles are at the 2 πr Asector of circle = ends of the box, median is a vertical line in the box, and P(A and B) = P(A)•P(B) independent 360 the max/min are at the ends of the whiskers. P(A and B) = P(A)•P(B/A) dependent 1 Asemicircle = π r 2 P(A or B) = P(A) + P(B) mutually exclusive Helpful in interpreting the distribution of data. 2 P(A or B) = P(A) + P(B) – P(A and B) not exclusive 1 P(B/A) = P(A and B)/P(A) conditional probability Aquarter circle = π r 2 4 P(B/A) means probability of B given A has occurred. Literal equations: Sets: Exponential Growth and Decay: A ∪ B Union - all elements in both sets. a = b + cd, solve for c. Decay: y = ab x where a > 0 and 0 < b < 1 a – b = cd A ∩ B Intersection - elements where sets overlap. a–b=c Growth: y = ab x where a > 0 and b > 1 d Use same strategies A ' Complement - elements not in the set. { } or ∅ means null set. as for solving equations. Copyright © Regents Exam Prep Center http://regentsprep.org ...
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