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1 Integrated Algebra Review Sheet Merritt Brown, Andy Janovsky and Millie Rivera-Perez This packet contains most of the information you will need to know in order to solve problems on the Integrated Algebra Regents Exam Equations and Inequalities 1. Evaluate expressions and formulas containing powers and parentheses and the four basic arithmetic operations. 2. Solve linear equations and inequalities and represent solution sets graphically. 3. Solve quadratic equations both algebraically and graphically. 4. Solve systems of linear equations and inequalities algebraically and graphically. 5. Solve systems consisting of a linear equation and a quadratic equation algebraically and graphically. 1 2 Triangles Area bh Right a 2 b2 c2 1 right angle Perimeter leg1 leg 2 hypotenuse 2 2 2 P= a + b + c Quadrilaterals Name Properties Parallelogram Two pairs of opposite sides Two pairs of opposite sides Rectangle Parallelogram with 4 right angles Rhombus Parallelogram with 4 sides. Square Trapezoid 4 sides and 4 right angles Exactly one pair of sides. Isosceles Trapezoid Trapezoid with exactly one pair of sides. Areas of irregular figures “ break them up” Surface area of a rectangular prism= 2lw 2lh 2wh Area and Perimeter Formula A=bh A=lw P = 2l + 2w 1 A d1d 2 2 A s2 P = 4s 1 A h(b1 b2 ) 2 Surface area of a cube= 6s 2 2 Surface area of a cylinder= 2 r 2 rh 2 Volume of a rectangular prism= length width height Volume of a cylinder = r 2 h Volume of a sphere = 4 3 r 3 Regular Polygon – perimeter = sum of all sides Properties Additive Identity : a + 0 = a Multiplicative Identity a 1 a Additive Inverses (opposites): 8 + (-8) = 0 Multiplicative Inverses (reciprocals) Commutative Addition: a + b = b + a Commutative Multiplication ab=ba Associative Addition (a + b) + c = a + (b + c) Associative Multiplication (ab)c=a(bc) a b 1 b a Distributive Multiplication over Addition: a( b + c) = ab + ac Circles C 2 R C D D 2R A R2 Proportion and Variation If a c b d then ad bc The product of the means is equal to the product of the extremes. If y varies directly as x then x1 x2 y1 y2 and y kx (k is a number ) . Line through (0,0). longest side small If two triangles are similar then longest side big shortest side small shortest side big Additional Facts Scientific notation 8.56 104 85,600 8.56 104 .000856 Consecutive Integers: n, n+1, n+2 Consecutive Odd or Even integers: n, n+2, n+4 Two numbers in the ratio 3:5 can be represented as 3x and 5x where x is a number. Coordinate Geometry slope change in y change in x slope y2 y1 x2 x1 Parallel lines have the same slope. Example: y = 5x +6 and y=5x – 9. 2 Asmall ssmall 2 Abig sbig 3 2 3 If two lines are perpendicular, the product of their slopes is –1. Example: y x 5 and y x 7 3 2 6 2 3 1 These are also called opposite reciprocals. 6 3 2 The slope of a horizontal line is zero. Example (y = 6) The slope of a vertical line is undefined. Example (x=5) The general equation for a straight line is y = mx + b where m is the slope and b is the y-intercept. The equation of a line in point slope form is y y1 m( x x1 ) where m is the slope and x1 , y1 are the given coordinates of a point on the line. Probability A sample space is all possible outcomes for an event. A tree diagram is one way to list all events in the sample space. Biased means UNFAIR! Theoretical probability is what should happen. Empirical probability is doing the experiment, recording results, and assigning probabilities. The counting principle states that if an event can occur x ways and a second event can occur y ways, then both events can occur in that order xy ways. Example: The school cafeteria has 4 types of entrees, 6 different drinks, and 3 choices of desserts. How many different lunches consisting of 1 entrée, 1 drink, and 1 dessert are possible? 4 63 72 different lunches The probability of a certain event = 1. The probability of and impossible event is 0 The theoretical probability of an event E…… P( E ) number of outcomes in event E number of outcomes in sample space The probability of “NOT A” = 1 P A P(A or B) = P(A) + P(B)- P(A and B) If events A and B are independent, then P(A and B)= P(A)P(B) Examples: On a single roll of a die what is the probability of rolling 1 1 5 Not a six P(not 6) 1 6 6 6 1 3 1 3 P 3 or odd (because 3is odd ) 6 6 6 6 A six P 6 A 3 or and odd number When calculating “without replacement” probabilities remember to decrease the denominator by 1 Example: If there are 8 red marbles and 2 two green marbles in a jar. Find the probability of drawing two red 8 8 64 8 7 56 marbles? With replacement P Two red Without replacement P Two red 10 10 100 10 9 90 4 Permutations are arrangements where order is important, such as medals in a race use n Pr or Examples: How many ways can 10 runners finish first second and third? How many arrangements in the word AZALEA? 6! 6 letters 3! 3 A ' s n! P 720 ways 10 3 120 arrangements Statistics Data can be classified as qualitative, how you feel about something or quantitative, numerical. The measures of central tendency are the mean, mode, median, and range (both high and low) The mean (average) of a set of numbers is the sum of the numbers divided by the number of numbers. The median of a set of numbers is the middle number when the numbers are arranged in order. If the number of numbers is even, average the two middle numbers. To find the median term in a data set: n 1 where n=total frequency, if the quotient is a “.5” then there are two 2 numbers that represent the median. ***If you are asked which is the best measure of central tendency it is the median because the mean can be corrupted by outliers*** The upper quartile is the median of the scores above the median. The lower quartile is the median of the scores below the median. If your grade is the 90th percentile, then your grade is higher than the grades of 90% of the students in the class. A percentile rank is found by multiplying the percent by the total frequency. The mode of a set of numbers is the number that has the greatest frequency. A set of numbers can have more than one mode. In grouped data with interval length greater than 1 mean, median, and mode cannot be found because you do not have the original data. You can find the modal interval, and the interval that contains the median. A scatter plot determines correlation between two independent variables. Correlation may be positive, rising line of best fit, negative falling line of best fit or no correlation at all, scattered points. Causation is determined if one variable forces the other to behave that way. Graphs: Line graph, Circle graph, Frequency Histogram, Cumulative Frequency Histogram, Scatter Plot, Stem and Leaf Plot, Box and Whisker Plot Numbers Natural numbers are the counting numbers 1, 2, 3, 4, 5… Whole numbers are the natural numbers plus 0. 0, 1, 2, 3, 4, 5… Integers consist of natural numbers, zero, and, negatives of natural numbers ... –4, -3, -2, -1, 0, 1, 2, 3 … The absolute value of an integer is the magnitude of the integer (distance from 0). Examples: 5 5 6 6 7 7 calculator: mathnum1 7 ENTER 5 Sample operations: 5 3 8 6 5 5 3 8 6 5 30 5 3 2 30 5 3 2 6 5 30 6 5 30 A prime numbers is a natural number greater than 1 that can only be divided by one and by itself. A composite number is a number greater than 1 that is not prime. Composite numbers have more than two factors. Example 6 has factors 1, 2, 3, and 6. The greatest common factor (gcf) of two numbers is the largest number that divides both numbers. Order of operations (parentheses, powers, divide and multiply left to right, subtract and add left to right. Exponents and radicals A rational number is a number that can be expressed as the ratio of two integers. When expressed in decimal form rational numbers are either repeating or terminating decimals. Examples: 4 5 6 7 8 9 5.4 .3333... 9 25 .3 49 .373737... .37 An irrational number cannot be expressed as a ratio of integers. Examples: 3 8 5 The set of real numbers consists of all rational and irrational numbers. Multiply radicals: 7 5 35 3 7 4 5 12 35 5 5 5 Dividing radicals: 18 35 6 7 3 5 Simplify radicals: 50 25 2 25 2 5 2 7 50 7 25 2 7 25 2 7 5 2 35 2 Like 3 56 5 9 5 10 7 7 9 7 Unlike Combining radicals: 7 12 5 3 7 4 3 5 3 7 4 3 5 3 7 2 3 5 3 14 3 5 3 19 3 6 Negative and Zero Exponents: 1 1 32 2 3 9 50 1 5 ^ 0Enter 1 61 6 3^ 2ENTER Math1 Enter 3 7 1 7 3 6 ^ 1( ENTER)(math1) ENTER 3/ 7 ^ 1 ENTER MATH1 ENTER When multiplying monomials, multiply coefficients and add exponents. Example: 5x 4 x 20 x 3 6 9 When multiplying polynomials, DISTRIBUTE and follow rules for multiplication above. 3x 2 2 x3 7 3x 2 x 3x 7 2 Monomial X Polynomial 3 2 6 x5 21x 2 Polynomial X Polynomial (foil) x 6 3x 1 x 3x x 1 6 3x 6 1 3 x 2 1x 18 x 6 3 x 2 17 x 6 When raising a power to a power, multiply the exponents and raise the base. Example: 4 x 2 (43 ) x 2 64 x 2 x3 64 x6 3 3 When dividing monomials, divide coefficients and subtract exponents. Example: 12 x10 3 x8 2 4x When dividing a polynomial by monomial, divide each term in the polynomial 9 x3 15 x 2 3x 9 x3 15 x 2 3x 3x 2 5 x 1 Example: 3x 3x 3x 3x When adding or subtracting polynomials, combine only like terms and operate only on the coefficients. Example: x 3x 6 3x 2 2 x 9 4 x2 x 3 2 7 3a 5 6a 2 5a 1 6a 2 2a 4 Equations and Inequalities First degree equations: Isolate the variable by using inverse operations 3x 4 14 2.5a 6 a 3 +4= +4 3 x 18 3 3 a 6 a 6 1.5a 9 1.5 1.5 x6 Literal equations: 3 3 x 9 21 4 x 9 21 3x 36 84 4 4 36 36 3 x 120 3 3 a6 Solve for x : ax b c b b ax c b a a c b x a x 40 Solve forc :e mc 2 e mc 2 m m e c2 m e c2 m e c m First degree inequalities: Same as equations BUT if you are dividing by a negative sign changes 3x 5 26 5 5 3 x 21 (reverse) 3 3 x 7 or solid dot Graph of the solution 7 < or > hollow dot Quadratics: All can be solved by factoring but ax 2 k is quicker by traditional solution. 3 x 2 192 3 3 x2 5x x 2 24 5 x x2 64 x 8 x2 5x 0 x 2 5 x 24 0 x x 5 0 x 8 x 3 0 x 8, 3 x 0,5 Answers are the roots of the equation, where the graph would cross the x-axis if the equation were graphed. 8 Systems of Equations Linear Solve for x and y 3x 2 y 5 2 x 4 y 18 Multiply to make opposites 2 x 4 y 18 = 2 x 4 y 18 Quad-Linear Solve algebraically for x and y: y 2 x 2 5x 5 and y x 5 Substitute value of y into second equation 2 3x 2 y 5 = 6 x 4 y 10 8 x 8 8 8 x 1 2 x2 5x 5 x 5 2 x2 6 x 5 5 5 5 2 2x 6x 0 2 2 x 3x 0 x x 3 0 x 0 and x 3 Find Y by substitution 3 1 2 y 5 3 2 y 5 +3 =+3 2y 8 2 2 y4 Solution x=-1 and y=4 or 1, 4 Find Y’s by substitution yx5 y0 5 y 3 5 y 5 y 8 Solutions 0,5 and 3,8 Exponential Growth and Decay: Decay: y ab x where a 0 and 0 b 1 A is the amount at beginning. B is rate of decay which is 100% minus given rate and X is the time. Example: Kathy plans to purchase a car that depreciates (loses value) at a rate of 14% per year. If the initial cost of the car is $21,000, what is the value of the car after 3 years? Let a= 21000 b= .86 ( 100% - 14%) and x= 3 v 21000 .86 21000 (.86) ^ 3 13357.176 So the car is worth $13,357.18 after 3 years. 3 Growth: The same formula except now B is found by adding the rate to 100% Example: Cassandra bought an antique dresser for $500. If the value of her dresser increases 6% annually, what will the value of the dresser be at the end of 3 years? Round your answer to the nearest dollar. Let a= $500 b= 1.06 (100% + 6%) and x= 3years 3 v 500 1.06 500 1.06 ^ 3 595.508 So to the nearest dollar it is worth $596.00 Special Products and Factoring Factoring Binomials: Greatest common factor (GCF): Find the largest number that divides into each coefficient. Take the lowest power of any common variable terms. Divide each term by GCF. Write answer as the product of GCF Answertodivision Example: Factor 15 x 4 10 x 2 y GCF= 5x 2 15 x 4 10 x 2 y 5 x 2 3x 2 2 y 5x2 5x2 Difference of squares (DOS): This binomial does not have a common factor. It factors to two binomials according to this rule: ( x2 y 2 ) x y x y Examples: x 9 x 3 x 3 25x 49 5x 7 5x 7 2 2 36a b 6a b 6a b 2 2 Factoring trinomials: Trinomials of the form x 2 bx c can be factored by answering one question. What two numbers multiply to “c” and add to “b”. If “c” is positive signs on the factors are the same: x2 7 x 12 x 4 x 3 x2 9 x 20 x 4 x 5 If “c” is negative signs on the factors are different, and sign on “b” tells you the sign of the “bigger” number. a2 a 30 a 6 a 5 a2 3a 28 a 7 a 4 . When factoring completely, first factor out the GCF then if a binomial remains factor a difference of squares. If a trinomial remains factor the trinomial accordingly. Examples: 3ax 2 48a 3a x 2 16 3a x 4 x 4 3x 2 15 x 42 3 x 2 5 x 14 3 x 7 x 2 Vocabulary and Symbols Words Symbols The sum of x and 5 x+5 x increased by 5 5 more than x The difference between x and 5 x decreased by 5 5 less than x, 5 fewer than x The product of 5 and x Multiply x by 5 The quotient of 5 divided by x 5 is greater than x 5 is more than x 5 is less than x 5 is less than or equal to x 5 is at most x 5 is greater than or equal to x 5 is at least x 5 is not equal to x x-5 5x 5 x 5>x 5<x 5 x 5 x 5 x Formulas for Some Problems distance=rate time rate= distance time time= distance rate Percent change = amountof change x100 starting number Discount Cost (%) Sale Pr ice Cost Discount Total cost = unit price x quantity sales tax = price x tax rate 9 10 Error in Measurement: Re lativeError measure actual actual Percent Error Re lativeError x100 Trigonometry tan( A) leg opposite A leg adjacent to A sin( A) leg opposite A hypotenuse cos( A) leg adjacent to A hypotenuse If you are given two sides and asked to calculate an angle, use tan 1 , cos 1 or sin 1 Graphs of linear functions and inequalities The general equation for a straight line is y = mx + b where m is the slope and b is the y-intercept. When the equation is in general form, (y = mx + b), you can graph it by hand or enter the equation into the y button on the calculator. Next press table for a list of x and y coordinates will appear. Choose at least three points, graph draw the line and label. Example: 4 x 2 y 8 -4x = -4x Slope= 2 up two, right one from the y intercept = (0,-4) 2 y 4 x 8 or 2 2 2 Press y then 2 X , T , , n - 4 enter then table Inequalities can graphed in the same manner with three exceptions: y 2x 4 When dividing by a negative the inequality symbol reverses. The graph of the line is dashed when < or > is being graphed. The solution is a shaded region, if or shade under the line or shade over the line Example y 2 x 6 +2x +2x y 2x 6 Y intercept ( 0 , -6) and slope= 2 up two right one draw a solid line and shade under. Mark shaded region solution set S Graphs of non-linear functions Parabola: y ax 2 bx c If a>0 the parabola holds water MIN. If a<0, the parabola looks like the path of a projectile MAX. If a is large the parabola is narrow. If a is small the parabola is wide. If b is increased the parabola shifts b units to the left, decreased shifts b units to the right. Changes in c result in a change in the yintercept, so the parabola will move up if c is increased and down if c is decreased. 11 b Equation of axis of symmetry: x Use this value of x as the middle value after you press table . 2a To find the vertex algebraically substitute x into equation of parabola to find y. To find vertex graphically examine table OR 2nd TRACE (3/ 4) To determine the roots of a parabola examine the graph, see where it crosses the x axis. Roots are not ordered pairs x, y they are a set of the X values where parabola crosses x-axis 1st x, 2 nd x Absolute Value y abs graph has a “V” shape. Exponential y ax graph has an almost “half parabola” shape, NEVER touches x-axis Additional Information A function is a relationship where each “x” corresponds to only one “y”. A relation is a set of ordered pairs ( x ,y) The relation 3,7 2,5 1, 1 3, 5 is a function because each x has only one y. The relation 3,9 2,6 5,11 2,8 is not a function because the 2 has two different y’s 6 and 8. The graph of a function will intersect a vertical line once. An algebraic expression is a math phrase made up of operation signs, numbers, and/or variables. 2a 2 7 a 3 is an expression. An equation is a math sentence which shows that two expressions are equal. 3x 7 2x 21 is an equation. Sets and set notation: X / 2 X 3 is read the set of x’s which is greater than -2 and less than or equal to 3. The set can be expressed in several ways { -1,0,1,2,3} is a list of the integral elements. It can also be graphed on a number line with a hollow dot on -2 and a solid dot on 3. 1.2 X / 2 X 3 is read 1.2 is an element of set of x’s which is greater than -2 and less than or equal 3 because 1.2 lies in that interval. The answer can also be expressed in interval notation as (2,3] . The universal set ( U )or domain is the set of all elements under consideration for a given example. A B is the union of sets A and B, which is all the elements in both sets A B is the intersection of sets A and B which are the elements in that are in both sets. A ' is the complement of set A, which are the elements of the universal set that are not in set A. A set with no elements is called the empty { } or null set. Use one or the other to express this. Examples: U 1, 2,3...10 A 1,3, 4,7B 2, 4,6,7,8C 5,9 A B 1, 2,3, 4,6,7,8 A B 4,7 A C B ' 1,3,5,9,10 Venn diagrams are pictorial representations of union, intersection, and complements of sets. Using units to solve problems: Convert 30 miles per hour to feet per second. 30 miles 1 hour 1 min 5280 ft 30 5280 ft ft 44 hour 60 min . 60 sec mile 3600 sec sec Algebraic Fractions An algebraic fraction is the quotient of two expressions. 3a 7 x 2 16 x4 12 x 16 is undefined when x=4 x4 A fraction is in simplest form when the numerator and denominator have a GCF of 1. To simplify a fraction divide both numerator and denominator by the GCF, polynomials require factoring. 2 A fraction is undefined when the denominator is equal to zero. Simplify 3a 2 / 3a a 3a 2 GCF= 3a 12a / 3a 4 12a Simplify x 2 16 x 4 x 4 x 4 x 4 x4 1 x 4 In order to add or subtract fractions you must have like denominators Subtract: 6z 4z 2z 5 5 5 Add ( x 3)(3) (2 x 1) 4 3x 9 8 x 4 11x 5 x 3 2x 1 LCD =12 4 3 4(3) 3 4 12 12 12 When multiplying fractions, simplify first by factoring, “any top with any bottom”, then multiply horizontally. 16 x 14 y 2 2 2 y 4 y Multiply 21y 8 x 3 1 3 Multiply x 4 x 4 3x 3x 3x x 2 16 2 x 4 x 7 x 12 x 4 x 3 x 4 x 3 When dividing fractions multiply by the reciprocal of the divisor, “KEEP, CHANGE, FLIP” Divide: x 5 x 1 6a x 1 30a 3 25a 2 30a 3 x2 6 x 5 30a 3 2 2 2 2 x 25 x 6 x 5 x 25 25a 25a 2 5 x 5 x 5 x 5 Calculator Suggestions (TI-84) 1. Use the (-) key for negative numbers and put the negative number in parentheses. Example: If x= -3 evaluate 5 x 2 4 x 8 5 3 4 3 8 = 41 2 2. Use MATH1 to change decimals to fraction and MATH 2 for vice versa. Add 1 2 9 1/ 2 2 / 5 .9 math1 enter 2 5 10 3. Use (^) for powers greater than 2. Example : 53 5 ^ 3 4. Use MATH NUM1 for absolute value items and graphs MATH PRB2 for permutations n Pr and PRB4 for factorials n! 12! 12 math PRB4 enter 479001600 6 P3 6 math PRB 2 3 enter 120 5 Use 6 Before performing any trigonometry (SOHCAHTOA) check MODE and set DEGREE Looking for side use SIN COS TAN looking for angle use Sin 1 Cos 1 Tan1 7 Before graphing any function press STATPLOT 2nd y and make sure PLOTS are OFF.