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Algebra 2

Algebra 2. Semester 2 Note Sheet. Make your power card. 2 2 = 4 3 2 = 9 4 2 = 16 5 2 = 25 6 2 = 36 2 3 = 8 3 3 = 27 4 3 = 64 5 3 = 125 6 3 = 216 2 4 = 16 3 4 = 81 4 4 = 256 5 4 = 625 6 4 = 1296 2 5 = 32 3 5 = 243 2 6 = 64 2 7 = 128

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Algebra 2

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  1. Algebra 2 Semester 2 Note Sheet

  2. Make your power card 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 23= 8 33 = 27 43 = 64 53 = 125 63 = 216 24= 16 34 = 81 44 = 256 54 = 625 64 = 1296 25 = 32 35 = 243 26 = 64 27 = 128 72 = 49 102 = 100 132 = 169 162 = 256 82 = 64 112 = 121 142 = 196 172 = 289 92= 81 122 = 144 152 = 225 182 = 324

  3. Definition of Rational Exponents For any nonzero number b and any integers m and n with n > 1, except when b < 0 and n is even

  4. NOTE: There are 3 different ways to write a rational exponent

  5. Let f(x)=3x1/3 & g(x)=2x1/3. Find (a) f(x) + g(x) (b) f(x) – g(x)(c) the domain for each. • 3x1/3 + 2x1/3 = 5x1/3 • 3x1/3 – 2x1/3 = x1/3 • Domain of (a) all real numbers Domain of (b) all real numbers

  6. Let f(x)=4x1/3 & g(x)=x1/2. Find (a) the product, (b) the quotient, and (c) the domain for each. • 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6 = 4x1/3-1/2 = 4x-1/6 = (c) Domain of (a) all reals ≥ 0, because you can’t take the 6th root of a negative number. Domain of (b) all reals > 0, because you can’t take the 6th root of a negative number and you can’t have a denominator of zero.

  7. Composition • f(g(x)) means you take the function g and plug it in for the x-values in the function f, then simplify. • g(f(x)) means you take the function f and plug it in for the x-values in the function g, then simplify.

  8. Let f(x)=2x-1 & g(x)=x2-1. Find (a) f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the domain of each. (a) 2(x2-1)-1 = (c) 2(2x-1)-1 = 2(2-1x) = (b) (2x-1)2-1 = 22x-2-1 = (d)Domain of (a) all reals except x=±1. Domain of (b) all reals except x=0. Domain of (c) all reals except x=0, because 2x-1 can’t have x=0.

  9. 6.5 Opener • Write the Parent equation for each of the • the graphed functions. y=x3 y=x y=x2 State the Domain and Range from each graph

  10. Parent Square Root

  11. Parent Cube Root

  12. Graphing using a(x – h) +k form • Step 1: graph a f(x) using 3 points • Step 2: translate those 3 points left or right, then up or down. Ex 1: Ex 2: Start at (h,k) and “a” acts as your slope

  13. Fundamental Counting Principle Two events If one event can occur in m ways and another event can occur in n ways, then the number of ways both events can occur is Three or more events The fundamental counting principle can be extended to three or more events. For example, if three events can occur in m, n and p ways, then the number of ways that all three events can occur is

  14. Counting principle with repetition Standard configuration of a license plate is 1 letter followed by 2 digits followed by 3 letters. a) How many different license plates are possible if letters and digits can be repeated? ___ ___ ___ ___ ___ ___ 26 • 10 • 10 • 26 • 26 • 26 = 45,697,600 plates are possible with repetition

  15. Counting principle with repetition Standard configuration of a license plate is 1 letter followed by 2 digits followed by 3 letters. b) How many different license plates are possible if letters and digits can NOT be repeated? ___ ___ ___ ___ ___ ___ 26 • 10 • 9 • 25 • 24 • 23 = 32,292,000 plates are possible without repetition

  16. Factorial The symbol for factorial is ! n! is defined where n is a positive integer as follows: n!=n•(n-1)•(n-2)•…•3•2•1 Example: 5!=5•4•3•2•1 The number of permutations of n distinct objects is n!

  17. Permutation of n objects taken r at a time The number of permutations of r objects taken from a group of n distinct objects is given by this formula Order matters!

  18. Back to our demo CD... What if I wanted to burn just 4 of my 10 songs onto a demo CD for my band – how many ways could I arrange the 4 songs? I have 10 songs and I want to arrange 4 of them on a CD, so this is a permutation where n=10 and r=4.

  19. Permutations with repetition The number of distinguishable permutations of n objects where one object is repeated time and another object is repeated times, and so on, is:

  20. Example Find the number of distinguishable permutations of the letters in MIAMI. There are 5 letters is MIAMI. M and I are each repeated 2 times. The number of distinct permutations is

  21. Combination -- a selection of r objects from a group of n objects when order is not important Examples: Choosing 3 flavors of ice cream out of 10 choices to put in a bowl Using 2 eggs out of a carton of 12 to mix into a cake mix Choosing 4 people out of a class of 32 to go get more chairs

  22. Combination The number of combinations of r objects taken from a group of n distinct objects is denoted by and is given by this formula:

  23. Theoretical probability If all outcomes for an event are equally likely, then the theoretical probability that event A will occur is: P(A) = number of outcomes in event A total number of outcomes The theoretical probability of an event is often just called the probability of the event yes total

  24. Example Roll a standard 6-sided die. What is the probability of rolling a five? There are 6 possible outcomes (total) Only 1 of the outcomes is a 5 (yes, that’s what I want) P(rolling a 5)=

  25. Odds When all outcomes are equally likely, the odds in favor of an event are: Odds in favor = number of outcomes in A of event A number of outcomes not in A yes no TOTAL = top + bottom

  26. Odds When all outcomes are equally likely, the odds against an event are: no yes Odds against = number of outcomes not in A an event A number of outcomes in A

  27. Overlapping EventsTwo events are overlapping if they have one or more outcomes in common.P(A or B) = P(A) + P(B) – P(A and B)

  28. Disjoint EventsTwo events are disjoint, or mutually exclusive, if they have no outcomes in common. P(A or B) = P(A) + P(B) B A

  29. Probability of the Complement of an Event • The event A, called the complement of event A, consists of all outcomes that are not in A. • The probability of the complement of A is

  30. Probability of Independent events • If A and B are independent events, then the probability that both A and B occur is: • P(A and B) = P(A)· P(B) • If there are more than 2 independent events, multiply all of the probabilities together.

  31. Example • A spinner has 6 equally sized sections of different colors. If the spinner is spun 4 times, what is the probability of landing on yellow (Y) each of the 4 times? • P(Y and Y and Y and Y) • = =

  32. Dependent Events • Two events are dependent if the occurrence of one affects the outcome of the other. • Example: If you are going to pick two marbles out of a bag and you give the first one to a friend before you pick the second one…you will have a different number of marbles in the bag for the second draw, which affects the probability.

  33. Conditional Probability • The probability that B will occur given that A has occurred. • This is written as P(B|A) • “the conditional probability of B given A”

  34. Probability of Dependent events • If And B are dependent events, then probability that both A and B occur is • P(A and B) = P(A) · P(B|A)

  35. Measures of central tendency • A number used to represent the middle or center of a set of data • Mean – the average – the sum of a set numbers divided by how many numbers are in the set. The symbol for mean is • Median – the middle number when the numbers are written in order (or the average of the two middle numbers) • Mode – the number or numbers that occur most frequently

  36. Measure of dispersion • A statistic that tells you how dispersed, or spread out a set of data is • Range – the difference between the highest and lowest data values • Standard deviation- describes the typical difference between a data value and the mean. The symbol for standard deviation is σ, which is read as “sigma.”

  37. Other data points • Lower Quartile – Q1 median of the lower half of the data • Upper Quartile - Q3 median of the upper half of the data • Outlier - A value that is much greater or much less than the other values in the set

  38. Use the data • To find the range: subtract the max and min values • To find the IQR (Inter Quartile Range): Subtract the values for Q3 and Q1 Range = Max – Min IQR = Q3 - Q1

  39. Calculator notes • Press stat, enter, then type the data values, pressing enter between each value (data will appear in L1) • Press stat, select calc, press enter – you will see the words 1-Var stats, press enter twice • Use the down arrow to read down the long list of stats

  40. Adding a constant to data values When a constant is added to every value in a data set, the following are true: The mean, median, and mode can be obtained by adding the same constant to the mean, median, and mode of the original data set. The range and standard deviation are unchanged.

  41. Multiplying data values by a constant When each value of a data set is multiplied by a constant, the new mean, median, mode, range, and standard deviation can be found by multiplying each original statistic by the same constant.

  42. Sampling Methods – ways to choose your sample • In a self-selected sample, members of a population can volunteer to be in the sample • In a systematic sample, a rule is used to select members of the population, such as every third person. • In a convenience sample, easy-to-reach members of a population are selected, such as those in the first row. • In a random sample, each member of the population has and equal chance of being selected.

  43. Margin of Error • The margin of error gives a limit on how much the responses of the sample would differ from the responses of the population. • For example, if 40% of the people in a poll prefer candidate A, and the margin of error is + 4%, then it is likely that between 36% and 44% of the entire population prefer candidate A.

  44. Margin of error formula • When a random sample of size n is taken from a large population, the margin of error is approximated by this formula: Margin of Error =

  45. Box and Whisker Plot • A box and whisker plot is a graphical display of the five number summary • Draw a scale to include the lowest and highest data values • Draw a box from Q1 to Q3 • Include a solid line through the box at the median • Draw solid lines, called whiskers from Q1 to the lowest value and from Q3 to the highest value.

  46. How to Compute Quartiles • Order the data from smallest to largest. • Find the median. This is the second quartile, Q2. • The first quartile Q1 is the median of the lower half of the data; that is, it is the median of the data falling below Q2, but not including Q2 • The third quartile Q3 is the median of the upper half of the data; that is, it is the median of the data falling above Q2 but not including Q2

  47. Interquartile Range • The interquartile range (IQR) is the difference between • Q3 and Q1 or Q3 –Q1 For our data set Q1 is 20, Q3 is 60, so the interquartile range is 60-20 = 40

  48. Five-Number Summary • Lowest Value or min • Q1 • Median (Q2) • Q3 • Highest value or max

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