Sol algebra 2 formula sheet, Cheat Sheet of Algebra

Download Sol algebra 2 formula sheet and more Cheat Sheet Algebra in PDF only on Docsity! 1 STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II 4th Nine Weeks, 2018-2019 2 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for students and parents. Each nine weeks’ Standards of Learning (SOLs) have been identified and a detailed explanation of the specific SOL is provided. Specific notes have also been included in this document to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models for solving various types of problems. A “ ” section has also been developed to provide students with the opportunity to solve similar problems and check their answers. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. The document is a compilation of information found in the Virginia Department of Education (VDOE) Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE information, Prentice Hall Textbook Series and resources have been used. Finally, information from various websites is included. The websites are listed with the information as it appears in the document. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. The Algebra II Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of questions per reporting category, and the corresponding SOLs. Standard Normal Probabilities Table entry Table entry for z is the area under the standard normal curve z to the left of z. z 00 01 .02 .03 04 .05 .06 07 08 .09 3.4 .0003 =.0003 = .0003 .0003 .0003 .0003 .0003 .0003 .0003 0002 3.3 .0005 0005 0005 0004 .0004 .0004 .0004 0004 0004 .0003 =3.2 0007 “0007 .0006 -0006 0006 .0006 .0006 .0005 .0005 0005 3.1 .0010 .0009 .0009 0009 + .0008 + .0008 .0008 .0008 .0007 .0007 3.0 .0013 .0013 = .0013 0012 0012 0011 = .0011 0011 .0010 0010 2.9 0019 .0018 .0018 .0017. 0016 00160015 0015 0014 = 0014 —2.8 .0026 8.0025 0024 0023 .0023 .0022 .0021 0021 .0020 .0019 27 «6.0035 = 0034 = .0033 .0032 «0031 = .0030) 0029 0028 )§= 0027) = 0026 2.6 0047 (0045 .0044 0043 0041 .0040 =«.0039 0038 §=§=.0037 0036 2.5 .0062 .0060 .0059 .0057. 0055 =.0054 «0052. 0051 9=.0049 60048 2.4 0082 0080 = =.0078 :0075 0073 .0071 .0069 .0068 .0066 0064 2.3 .0107 .0104 .0102 0099 .0096 .0094 .0091 .0089 0087 .0084 —2.2 0139 .0136 = .0132 0129 0125 .0122. 0119 0116 = .0113 0110 2.1 0179 0174 = .0170 0166 .0162 .0158 .0154 .0150 .0146 .0143 2.0 .0228 .0222 .0217 .0212. 0207 = .0202, 0197) 0192 «0188 = 0183 “1.9 0287 0281 .0274 0268 .0262 .0256 .0250 0244 .0239 .0233 -1.8 .0359 .0351 0344 (0336 «=6.0329 0322) .0314 = .0307— 0301 += .0294 -17 «4.0446 = 04360427 0418 .0409 «0401 = 03920384 = 0375 0367 =1.6 .0548 .0537 = .0526 -0516 0505 .0495 0485 0475 .0465 0455 -1.5 .0668 .0655 .0643 0630 .0618 .0606 .0594 .0582 .0571 .0559 =1.4 .0808 .0793 .0778 -0764 0749 0735) .0721 .0708 .0694 0681 -1.3 .0968 .0951 .0934 0918 .0901 .0885 .0869 .0853 .0838 .0823 —12 41514131 .1112 1093 1075) .1056 = 61038 )= 1020) 1003S .0985 -1.100 1357) 1335.14 129200 1271 0.1251 .12300 1210 190.1170 -1.0 .1587 1562 .1539 1515 1492 1469 «1446014231401 1379 0.9 1841 .1814 .1788 17620 «1736017110 .1685 = 166009 1635-1611 0.8 .2119 2090 = .2061 2033 2005 1977) 19491922 .1894 1867 0.7 2420 2389 .2358 2327 229622662236) 2206) 21772148 0.6 .2743 2709 = .2676 2643 2611 2578 =«.2546 = 25142483 2451 0.5 3085 3050 3015 2981 «2946 2912) 2877) 2843028102776 0.4 3446 «34093372 333606 3300) 63264) .3228 03192) 3156 3121 0.3 3821 3783) .3745 -3707- 3669) 36320 635943557) 03520-3483 0.2 4207 4168 = .4129 4090 4052 4013 3974 3936) 3897 3859 0.1 4602 4562 4522 4483 «44443044040 43640 4325042864247 0.0 .5000 4960 = .4920 -4880 4840 «= 48014761 4721 4681 4641 Standard Normal Probabilities Table entry Table entry for z is the area under the standard nermal curve to the left of z. Zz 00 01 02 03 04 05 06 07 08 09 0.0 -5000 -5040 .5080 -5120 -5160 5199 5239 = 5279 5319 5359 O21 5398 .5438 5478 .5517 5557 .5596 .5636 5675 5714 .5753 0.2 5793 5832 5871 -5910 5948 5987 6026 6064 6103 6141 0.3 -6179 6217 .6255 -6293 .6331 .6368 .6406 6443 .6480 .6517 0.4 .6554 .6591 .6628 .6664 6700 .6736 .6772 .6808 6844 .6879 0.5 6915 .6950 .6985 7019 7054 «6.7088 = 71237157 — 7190 7224 0.6 7257 7291 7324 7357 7389 7422 74547486 7517 7549 07 7580 .7611 .7642 8.7673) 770477347764 L7794 £78.23 7852, 0.8 7881 8.7910 8.7939 -7967 7995 .8023 .8051 8078 .8106 8133 0.9) ©6.8159. 8186 «= «82128238 = 82640 8289 )=— «8315 «8340 = «8365 8389 1.0 8413 .8438 8461 .8485 .8508 8531 8554 8577 8599 8621 Li 8643 .8665 .8686 -8708 «864.8729 «4.8749 8770) 8790 )~— 8810 8830 12 .8849 .8869 .8888 .8907 8925 .8944 8962 8980 8997 9015 1.3 9032 .9049 = .9066 9082 «9099 9115) .9131 9147 = .9162 3177 1.4 9192 “9207 = .9222 -9236 9251 9265 .9279 .9292 9306 9319 15 9332 9345 .9357 .9370 .9382 .9394 9406 9418 9429 9441 1.6 9452 9463 9474 9484 -9495 -9505 .9515 -9525 -9535 .9545 17 6.9554 9564 69573 958209591) 6959969608 §=—.9616 = 9625 9633 1.8 9641 .9649 .9656 9664 .9671 .9678 .9686 .9693 .9699 9706 19 9713 9719 .9726 9732. 9738) .9744. 9750) 97569761 9767 2.0 9772 .9778 9783 «86.9788 «=.9793) 9798) = 69803) = 69808 «= 9812817 2.1 9821 .9826 .9830 9834 84.9838 86.9842) .9846)0=— 9850) 0S 9854 9857 22 861 .9864 .9868 “9871 ‘9875 9878 .9881 9884 9887 9890 2.3 9893 .9896 .9898 -9901 9904 §=6.9906 §=6.9909 9911 = 9913 9916 24 -9918 “9920 .9922 9925 9927 9929 = .9931 9932, 9934 9936 2.5 9938 8.9940 .9941 .9943 9945 9946 9948 9949 9951 9952 2.6 -9953 -9955 9956 “9957 -9959 -9960 -9961 9962 9963 .9964 27 965 .9966 .9967 9968 8.9969 .9970 9971 9972 .9973 9974 2.8 -9974 “9975 9976 “9977 ‘9977 9978 .9979 .9979 9980 9981 29 9981 .9982 .9982 9983 9984 «§©6.9984 9985 9985) 9986. 9986 3.0 9987 -9987 9987 -9988 -9988 989 .9989 .9989 -9990 .9990 3.1 9990 .9991 9991 9991 9992 9992 9992 .9992 .9993 9993, 3.2 -9993 9993 .9994 -9994 -9994 -9994 §=.9994 §=.9995 -9995 9995 3.3 9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997 3.4 9997 “9997 9997 9997 9997 9997 9997 ‘9997 9997 9998, 7 Sequences and Series AII.5 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, determining the nth term, and evaluating summation formulas. Notation will include  and an. Arithmetic Sequence Geometric Sequence A sequence where the difference between consecutive terms is a constant. (You add or subtract a constant value) A sequence where the difference between consecutive terms is a common ratio. (You multiply or divide a constant value) Examples: 3, 5, 7, 9, 11… (constant is +2) 25, 20, 15, 10… (constant is −5) Examples: 3, 6, 12, 24, 48… (ratio is 2 1 ) 6, 9, 13.5, 20.25, 30.375… (ratio is 3 2 ) Formula 𝑎𝑛 = 𝑎 + (𝑛 − 1)𝑑 𝑎 is the starting value, 𝑑 is the common difference and 𝑛 is the number of terms. Formula 𝑎𝑛 = 𝑎 ∙ 𝑟𝑛−1 𝑎 is the starting value, 𝑟 is the common ratio and 𝑛 is the number of terms. Example 1: What is the 35th term of the arithmetic sequence that begins 7, 4… 𝑎𝑛 = 𝑎 + (𝑛 − 1)𝑑 𝑎𝑛 = 7 + (35 − 1)(−3) 𝑎𝑛 = −95 Example 2: What is the 20th term of the geometric sequence that begins 1, 2, 4… 𝑎𝑛 = 𝑎 ∙ 𝑟𝑛−1 𝑎𝑛 = 1 ∙ 220−1 𝑎𝑛 = 524,288 Example 3: What is the missing term in this geometric sequence 9, ⎕ , 1 … 𝑎𝑛 = 𝑎 ∙ 𝑟𝑛−1 1 = 9 ∙ 𝑟3−1 1 9 = 𝑟2 𝑟 = 1 3 The missing term is 9 ∙ 1 3 = 3 Substitute your values (𝑎 = 7, 𝑛 = 35, 𝑑 = −3) Simplify Substitute your values (𝑎 = 1, 𝑟 = 2, 𝑛 = 20) Simplify Substitute your values (𝑎𝑛 = 1, 𝑎 = 9, 𝑛 = 3) Simplify Solve for the common ratio, 𝑟. 10 Example 2: In the finals of the diving meet referenced in Example 1, the top 3 finishers score points for their team. First place receives 10 points, 2nd place receives 8 points, and 3rd place receives 6 points. In how many ways can the 8 finalists finish in the top 3? Now, the order is important because 1st place gets more points than 2nd place. We will use a permutation! 𝑃 38 = 8! (8 − 3)! = 8! 5! = 40320 120 = 336 There are 336 possible ways that the top 8 divers can finish in the top 3. Statistics 1. A teacher is making a multiple choice quiz. She wants to give each student the same questions, but have each student's questions appear in a different order. If there are twenty-seven students in the class, what is the least number of questions the quiz must contain? 2. A coach must choose five starters from a team of 12 players. How many different ways can the coach choose the starters? Standard Deviation AII.11 The student will a) identify and describe properties of a normal distribution; b) interpret and compare z-scores for normally distributed data; and c) apply properties of normal distributions to determine probabilities associated with areas under the standard normal curve. Standard Deviation and Variance The standard deviation of a data set tells us how “spread out” the data is, if the data is very spread out, the standard deviation will be higher than if the data is all clumped together. The variance is another measure of how spread out the data is. Standard deviation is represented by σ (lowercase Greek letter sigma). The variance is just the standard deviation squared, σ². There is a way to calculate these values in the graphing calculator. 11 Example 1: The height in inches of the Washington Wizards starting lineup is shown below. Find the standard deviation and the variance of the data, round your answer to the nearest hundredth. 75, 80, 76, 79, 81 Start by entering the data into L1 in your STAT menu. We want to use the standard deviation that is represented by σ, therefore our standard deviation is 2.32.The variance is just the standard deviation squared = (2.32)² = 5.38 Example 2: Using the data from Example 1, how many of the starting lineups’ heights are within one standard deviation of the mean? The heights were 75, 80, 76, 79, 81 This question is referring to players who are both one standard deviation above the mean and one standard deviation below the mean. The mean was 78.2 inches, and the standard deviation was 2.32 inches. 78.2 + 2.32 = 80.52 inches 78.2 – 2.32 = 75.88 inches Then go to STAT, scroll over to CALC, and select 1: 1-Var Stats When you press ENTER twice, your calculator will display the single variable statistics. MEAN SUM of the DATA SUM Squared Sample Standard Deviation Population Standard Deviation Sample Size KEY Standard Deviation = 2.32 inches Variance = 5.38 inches 12 There is one player (81”) who is taller than one standard deviation above the mean and one player (75”) who is shorter than one standard deviation below the mean. This means that 3 players (80”, 76”, and 79”) are all within one standard deviation of the mean. Example 3: How short would a player have to be in order to be 2.5 standard deviations below the mean? First we need to calculate how many inches is 2.5 standard deviations. We can do this by multiplying the standard deviation by 2.5. 2.32 ∙ 2.5 = 5.8 𝑖𝑛𝑐ℎ𝑒𝑠 We can then subtract 5.8” from the mean of 78.2”. 78.2 − 5.8 = 72.4 𝑖𝑛𝑐ℎ𝑒𝑠 A player would have to be 72.4” tall to be 2.5 standard deviations below the mean. Standard Deviation Use the speeds of the top 10 fastest roller coasters, provided in mph, to answer the questions below. 128, 120, 107, 100, 100, 95, 93, 85, 85, 82 1. What is the standard deviation? (round to nearest hundredth) 2. What is the variance? (round to nearest hundredth) 3. How many coasters are within 1.25 standard deviations of the mean? 4. How fast would a coaster have to be going to be 3 standard deviations above the mean? Z-Scores A z-score tells us how many standard deviations a specific data point is from the mean. Z-scores can be positive or negative. If a z-score is positive it indicates that the data point is that many standard deviations above the mean, if a z-score is negative it indicates that the data point is that many standard deviations below the mean. If a data point has a z-score of zero, then that data point is the same as the mean of the data. Scan this QR code to go to a video tutorial on standard deviation. 15 Example 7: The graph below shows how temperatures were normally distributed across the globe one day last year. If 1,000 cities were sampled, how many cities had temperatures between 4° and 100°? Z-Tables When z-scores fall on an integer, it is easy to use the 68-95-99.7 rule to determine the cumulative probability of a range of values. Sometimes, you may need to use a z-table to determine the probability based on a z-score when the z-score is in decimal form. In order to use a z-table, follow the column on the left to find your ones and tenths place of your z-score. Then, follow the row at the top to find your hundredths place of your z-score. Find where they meet. This value represents the cumulative probability or area under the curve to the left of the z-score. 4° is 2 standard deviations below the mean and 100° is 2 standard deviations above the mean. 95% of the data points fall within 2 standard deviations of the mean, therefore 95% of the cities sampled would have had temperatures between 4° and 100°. 0.95 ∙ 1000 = 950 𝑐𝑖𝑡𝑖𝑒𝑠 16 Example 1: Use the z-table to determine the probability that a data value will fall below a data value associated with a z-score of 0.53. Example 2: The length of the life of an instrument is 12 years with a standard deviation of 4 years. Out of 500 instruments, how many can be expected to last 15 or more years? First, the z-score of a length of 15 must be found. 𝑧 − 𝑠𝑐𝑜𝑟𝑒 = 𝐷𝑎𝑡𝑎 𝑃𝑜𝑖𝑛𝑡 − 𝑀𝐸𝐴𝑁 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 15 − 12 4 = 0.75 Next, use the z-table above to find the probability value of a z-score of 0.75. You should find 0.7734 (or 77.34%). Remember, subtract this value from 1, because we are looking for the area under the curve to the right since we were asked about 15 or more years! 1 − 0.7734 = 0.2266 Multiply 0.2266 by 500. 0.2266 ∙ 500 = 113.3 ≈ 113 113 instruments represents 22.66% of 500 instruments. Look at what the value represents. Remember, that this value represents area to the left of the curve! If you were asked to find probability of values “greater than” the value associated with 0.53, you would have to subtract the value from 1. 1 - .7019 = 0.2981 Therefore, the probability or area under the curve is 0.7019 ( or 70.19%). 17 Standard Deviation The heights of the tallest 7 men ever recorded are shown below (in inches). Use these to answer the questions. 107, 105, 103.5, 99, 99, 99, 98 5. What is the z-score for 99 inches? 6. What is the z-score for 107 inches? 7. The tallest woman ever confirmed would have had a z-score of -1.13. How tall was she? 8. The means and standard deviations for two schools’ SAT scores is shown below. The z-score for the 95th percentile is 1.598. By how many points do the 95th percentile scores differ for each school? (Round to the nearest whole number.) School A: Mean = 1520, Standard Deviation = 110 School B: Mean = 1490, Standard Deviation = 155 9. The height of the men in the United States is normally distributed as shown in the graph. The mean is 69.25” with a standard deviation of 2.5”. What percent of the heights are between 66.75” and 74.25”? 10. The length of time that people can hold their breath under water is normally distributed with a mean of 32 seconds and a standard deviation of 12 seconds. Out of 750 people, about how many people would be expected to hold their breath for 42 seconds or longer? For 35 seconds or less? (Hint: Use the part of the z-table shown on page 15.)