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Properties Commutative: Change Order Reflexive: a=a (yes!) Associative: Regroup-move ( ) Symmetric: If a=b, then b=a Distributive: multiply through ( ) Transitive: If a=b and b=c then a=c Identities: 0 (Add) & 1 (Multiply) Substitution: replacing one term for two Inverse: Opposites (add #s to 0 ) & Reciprocals (multiply #s to 1) Algebra II SOL Study Sheet Quadratics Factoring Standard : ax2+bx+c ax2+bx+c Standard Form: Put in y= see where the graph crosses the x axis 2nd calc, Zero, left, right, enter Roots Zeros Solutions Factors (inverse #s of zeros) Monomials Coefficient, Base, Exponent Ex: 3x2 Rules of Exponents 1. an am =an+m Multiply: Add exponents 2. an /am =an-m Divide: Subtract exponents 3. (an)m =an*m Power of Power: Multiply exponents 4. a0 =1 Anything to the 0 a-n =1/an If an exponent is power is 1 5. negative upstairs then it will be positive in the downstairs. 1. Find a GCF (If there is one) 1. 2. Find a number that multiplies to give you c but adds to give you b. *c +b Write in ( x+ ) ( x+ ) Example of Case #1 x 2 16 x 64 = (x-8)(x-8) Example of Case #2 (you can use slide & divide) 3x 2 x 2 (3x -2)(x+1) 2 Difference of Squares: x 4 ( x 2)( x 2) Sum of Cubes: a3 b3 (a b)(a 2 ab b 2 ) Remember: Same Opposite Always Positive Difference of Cubes: a3 b3 (a b)(a 2 ab b 2 ) SOAP Solving Quadratic Equations ax 2 bx c 0 Remember: Roots = zeros = solutions = x-intercepts 1. Method 1: Solving by factoring b 2 4ac >0 a. If Factorable, factor the polynomial b. Set each factor equal to zero and then solve for x b 2 4ac <0 2. Method 2: Quadratic formula if polynomial is not factorable 2 real roots 2 imaginary roots b b 2 4ac 2 1 real root b 2 4ac =0 a. X = , b 4ac is called the discriminate (Duplicity) 2a MATRICES 3. Method 3: Type quadratic equation in calculator y = a. Graph the equation and where the parabola crosses the x axis is where the solution is b. Using the table function: i. 2nd table (2nd graph) : The solution is the x value when y is equal to 0 Absolute Value Equations and Inequalities |x+2|=7 (x + 2) = 7 or (x + 2) = - 7 x+2=7 x=5 x + 2 = -7 x = -9 x = -9 or or | 2x + 3 | > 6 Put is above 6 and below -6 2x + 3 < 6 [this is the pattern for "greater than"] 2x+3< -6 OR 2x+3 > +6 or x = 5 Remember: Greater - greatOR | 2x + 3 | < 6 Put it between -6 and 6 –6 < 2x + 3 < 6 [this is the pattern for "less than"] –6 – 3 < 2x + 3 – 3 < 6 – 3 –9 < 2x < 3 –9 /2 < x < 3/2 Then the solution to | 2x Between -9/2 and 3/2 ● Value is Included ○ Value is Not included –9 /2 < x < 3/2. Basic Graphing Concepts Domain: set of all x values Range: set of all y values X intercepts: where the graph crosses the x axis Y intercepts: where the graph crosses the y axis X < -9/2 or x > 3/2 Direct, Inverse, and Joint Variation Equations Direct Variation: y = kx, where k is a nonzero constant of variation k Inverse Variation: y = or xy = k where k 0 x Joint Variation: y = kxz Radicals and Radical equations 3 g(f(x)) = g(2x+3) = 2x + 3 + 2 = 2x + 5 + 3 | >6 is x> 3/2 OR X < -9/2 Then the solution to | 2x + 3 | < 6 is the interval Compositions of functions f(x) = 2x + 3 g(x) = x + 2 g(f(x)) substitute value of f(x) everywhere there is an x in the g function > 8x 8 = 2 x 2 3 x 2 3 1) Take the 8 =2 2) Look at variable x - See how many times 3 goes into 8 and how much is left over a b x x 3 5 power root x 5 x3 Solving Radical Equations 1) Isolate the radical 2) Square both sides 3) Solve for x 4) Check answer 2 x 3 17 2 x 14 ( 2 x ) 2 142 2 x 196 x 98 Arithmetic and Geometric Sequences 1) Arithmetic sequences: Add a common difference to each term to obtain the next term Formulas: an an 1 d an a1 (n 1)d Remember a n is the nth term a1 is the first term n is the number of the terms d is the common difference Arithmetic and Geometric Series 2) Geometric Sequences: Multiply the common ratio to each term to obtain the next term Formulas: an an 1 r an a1r n 1 Remember a n is the nth term a1 is the first term n is the number of the terms r is the common ratio REMEMBER SERIES = SUM OF TERMS IN THE SEQUENCE 1 is the lower limit and 6 is the upper limit Calculator steps: 1)2nd stat Math 5 sum(enter) 2) 2nd Stat OPS 5 seq(enter) 3) ( 2^n , x , 1 , 6) On calculator screen: Sum(seq(2^x, x , 1 , 6) = 126 Sum of the first six numbers is 126 Math, # 0, Summation, enter values Scatter Plots, Curve of Best Fit and predictions Scatter plots: graph of a real-life data Curve of Best fit: model the data in a scatter plot L1: Week L2: Weight in grams of radioactive material Look at graph to check the shape of the graph Choose a regression model that fits the shape. Solve: A chemist has a 100-gram sample of radioactive material. He records the amount of radioactive material every week for 6 weeks and obtains the data. 1. Read question to see what you are solving for and which data set should be x 2. We are solving for weight in grams 3. Turn on stat plots 4. STAT, EDIT,enter L1, L2 5. Stat Calc 4 0: ExpReg 6. Xlist: L1 7. YList: L2 (skip Freqlist) 8. STORE Vars y vars #1 #1 enter 9. Graph by hitting zoom 9 for zoomstat 10. Check window 11. What about Week 10? 12. 2nd, table set, 10, enter, 13. Table screen, next to week 10, 26.961 grams Combination or Permutation STATS Algebra 2 Normal Distribution: = mean ( average) – standard deviation (how far above or below you are from the mean) Permutation: # of ways you can select some items out of a group total, order DOES matter n! n Pr (n r )! Example: Selecting a president, vice president, historian and treasurer out of 15 members. 15 Z- score: The z-score a particular data point is the measure of how many standard deviations that point is away from the mean. You can calculate the z-score of any data point using the data point formula. x z x data point mean standard deviation Ex: The mean salary for an entry level engineer is $32,000 per year with a standard deviation of $2,500. Find the z-score of an engineer who is making $35000. 35000 32000 z 1.2 (1.2 standard deviations from the mean) 2500 Z-score and Probability/percent You can calculate the probability/percentage of an event occurring by using the z-score. P4 15! (15 4)! Math, PROB, # 2 nPr, enter values Combination - # of ways you can select some items out of a group total, order DOESN’T matter n! n Cr (n r )! r ! Example: selecting a committee of 4 members out of the total of 15 people where everybody has equal standing 15 C4 15! (15 4)!4! Ex: In a large group of guayule plants, the heights of the plants are normally distributed with the mean of 12 inches and standard deviation of 2 inches. Find the percent of the plants are shorter than 16 inches. z 16 12 2.0 2 Now look at the z-score table to determine what the percentage is 0.9777 -- probability 97.77% -- percentage Math, PROB, # 2 nCr,enter values