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Pre-AP Algebra II Math Team Summer Reading for 2021-2022

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Pre-AP Algebra II Math Team Summer Reading for 2021-2022 Pre-AP Algebra II Math Team Summer Reading for 2021-2022 Just a little love from Mr. Hurry to you. 1 Table of Contents FORMULAS and DEFINITIONS for MATH TEAM .................. 3 Powers Quizzes .................................................................................. 3 Miscellaneous Algebraic Laws, Rules, Formulas, and Facts ............... 4 Factoring ............................................................................................... 7 Factoring out a greatest common monomial .......................................................... 7 Factoring a difference of two squares ................................................................... 7 Factoring a perfect square trinomials ................................................................... 7 Factoring a trinomial with a lead coefficient of “1” .................................................. 7 Trigonometric Identities and Laws .................................................... 8 Law of Sines & Cosines ................................................................................ 8 Fundamental Identities ............................................................................... 8 Math Team related FORMULAS and DEFINITIONS ........................... 9 Conic Sections (Circles & Parabolas) ................................................. 12 Circles ....................................................................................................... 12 Parabolas .................................................................................................. 13 Graphing “12 Basic Functions” .......................................................... 14 Quiz #1 - Summer Reading Packet ..................................................... 15 2 FORMULAS and DEFINITIONS for MATH TEAM Powers Quizzes Please memorize all of the values listed directly below. You will need to know these by heart without using a pencil to calculate them. You will be given timed tests over these values beginning the first week of school. If you are really fast, you can actually get bonus points (you may need them!). BTW, Flash cards are a great idea. If you are able to complete a quiz like below in under 35 seconds you will be labeled for all eternity as “The Daddy” and will go down in math lore with high honors. If you are able to complete the quiz in under 30 seconds you will be mentioned in the same sentence as “The Great One,” Hamilton Simpson, who not only helped put Homewood on the map as a great math team school, but who also holds the record at 27 seconds. 2 24 392 42 16 52 25 3 28 33 27 43 64 53 125 4 2 16 34 81 44 256 54 625 5 2 32 35 243 45 1024 55 3125 6 2 64 36 729 46 4096 56 15625 27 128 37 2187 28 256 38 6561 29 512 210 1024 211 2048 212 4096 Here is what a sample “powers quiz” will look like. I would practice these this summer. If you can correctly finish a quiz like this in under 2 minutes you’ll be doing A or better work on the quiz. Name______________________________________________ Period_____________ POWERS QUIZ Math Team 1. 212 2. 45 3. 28 4. 54 5. 38 6. 4 6 7. 2 7 8. 2 6 9. 55 10. 56 11. 35 12. 34 13. 210 14. 53 15. 4 4 16. 211 17. 36 18. 37 19. 2 9 20. 25 3 Miscellaneous Algebraic Laws, Rules, Formulas, and Facts 1. You will also need to have memorized the following values… 112 72 49 132 169 192 361 0! 1 242 82 64 142 196 202 400 1! 1 392 92 81 152 225 212 441 2! 1 2 2 42 16 102 100 162 256 222 484 3! 1 2 3 6 52 25 112 121 172 289 232 529 4! 1 2 3 4 24 62 36 122 144 182 324 242 576 5! 1 2 3 4 5 120 252 625 2. factorials n! n n 1 n 2 n 3 2 1 3. double factorial n!! n n 2 n 4 n 6 4 2 when n is even n!! n n 2 n 4 n 6 3 1 when n is odd For example: 7!! 7 5 3 1 105 6!! 6 4 2 48 yy 4. Slope definition: m 21 Slope is the “rate of change” of a line. xx21 A If Ax By C, then slope, m 5. Slope (of a line) B If y mx b, then slope m 6. Point-slope form (of a line) y y11 m x x 2 For example: Determine the equation of a line going through the point 2, 3 having a slope of . 3 2 yx 32 3 2 yx32 3 7. Laws of Exponents (multiplication): m n am a n a m n ab a m b m a m a mn Examples: 4 5 a3 a 5 a 8 ab a 4 b 4 a 3 a 15 4 8. Laws of Exponents (division): mmm amn 1 a a na n m m a a b b Examples: 5 a 5 12 7 aa 6 a12 aa6 6 11 bb = aa12 5 7 9. Laws of Exponents, continued... 1 11 aa0 1 n Examples: x01 x 4 or x 4 an xx44 10. Know how to use the Quadratic Formula if ax2 bx c 0 b b2 4 ac x 2a Example: 2 Solve: 7xx 3 5 0 2 3 3 4 7 5 3 9 140 3 149 x 2 7 14 14 11. Relatively prime - two integers are relatively prime if their greatest common factor is 1. Example: Even though 8 and 9 are not prime, they are considered relatively prime because their greatest common factor is 1. a bi 12. Complex Numbers a real part b= imaginary part 54 i 5 2i 3 Ex) real part is 5 Ex) real part is 5 imaginary part is 4 imaginary part is 2 3 13. Pythagorean Theorem c2 a 2 b 2 22 14. Distance formula (plane) D x2 x 1 y 2 y 1 5 x1 x 2 y 1 y 2 15. Midpoint formula (plane) xymm,, 22 16. Discriminant - the discriminant is the value b2 4 ac coming from the equation ax2 bx c 0. If b2 4 ac >0 then there are two unequal real solutions to the equation If b2 4 ac =0 then there is a real double root to the equation If b2 4 ac <0 then there are two conjugate imaginary roots to the equation Ex) Question: Determine what values of k will give the following equation imaginary roots: 3x2 4 x 2 k 0 Ans: Based on the equation above, a3, b 4, and c 2 k . Therefore, since the equation is supposed to give imaginary roots… b2 40 ac 42 4 3 2k 0 16 24k 0 24k 16 2 k 3 17. Harmonic mean: a number , mh , is the harmonic mean of x and y if 1 11 2 1 1 is the average of and , that is mh xy mh x y An easier way to think about the harmonic mean of two numbers x and y can be found by 2xy 2tt13 mh or t2 xy tt13 This is essentially saying … “two times the product divided by the sum,” ex.) the harmonic mean of 6 and 12 is . 2 6 12 2 6 12 8 6 12 18 nn 1 18. Formula for the Sum of the first n numbers 1 2 3 n 2 Ex) Find the sum of 1 2 3 4 ... 50 or Find the sum of the first 50 terms. nn 1 50 51 Solution: 1275 22 6 n n1 2 n 1 19. Formula for the Sum of the first n squares 12 2 2 3 2 n 2 6 Ex) Find the sum of 12 2 2 3 2 4 2 ... 50 2 or Find the sum of the first 50 squares. n n1 2 n 1 50 51 101 Solution: 42925 66 nn2 12 20. Formula for the Sum of the first n cubes 13 2 3 3 3 n 3 4 Ex) Find the sum of 13 2 3 3 3 4 3 ... 50 3 or Find the sum of the first 50 cubes. nn22 122 50 51 Solution: 1625625 44 Factoring Factoring out a greatest common monomial Ex.) 3xy 12 x2 4x3 6 xy 10 x Factoring a difference of two squares Ex.) x2 25 4xy22 81 2a32 72 ab Factoring a perfect square trinomials Ex.) xx2 10 25 xx2 69 4xx2 12 9 Factoring a trinomial with a lead coefficient of “1” Ex.) xx2 9 14 xx2 4 12 xx2 89 7 Trigonometric Identities and Laws Law of Sines & Cosines Law of sines Law of cosines a2 b 2 c 2 2 bc cos A sinABC sin sin b2 a 2 c 2 2 ac cos B a b c c2 a 2 b 2 2 ab cos C Fundamental Identities Trigonometric Functions Reciprocal Identities Quotient identities y opp sin 1 r hyp sin sin csc tan x adj 1 cos cos cos r hyp sec y opp 1 cos tan tan cot x adj cot sin x adj 1 cot cot y opp tan r hyp 1 sec sec x adj cos r hyp 1 csc csc y opp sin 8 Math Team related FORMULAS and DEFINITIONS 1. Sum of the roots of a polynomial . axn bx n12 cx n yx z 0 b2 nd coefficient rr or 12 a1 st coefficient 2. Product of the roots of a polynomial . axn bx n12 cx n yx z 0 z constant or , if highest power is even a first coeff. rr12 z constant or , if highest power is odd a firstcoeff 3. Sum of the squares of the roots of . axn bx n12 cx n yx z 0 b2 2 ac r2 r 2 r 2 1 2 3 a2 Example for #1-3: For the following polynomial, f x 3 x5 7 x 4 5 x 3 12 x 2 6 x 9, find the . Sum of the roots: Product of the roots: b2 nd coefficient 7 7 constant 9 odd power, so . 3 a1 st coefficient 3 3 first coefficient 3 Sum of the squares of the roots: n n1 n 2 n 3 If ax bx cx dx 0 2 b2 2 ac 7 2 3 5 19 Then a2 32 9 4.
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