Algebra II Notebook
Wasatch High School
2011-2012
Student Name ______
Teacher Name ______
Algebra 2
Standard I: Students will use the language and operations of algebra to evaluate, analyze and solve problems.
Objective 1: Evaluate, analyze, and solve mathematical situations using algebraic properties and symbols.
a. Solve and graph first-degree absolute value equations of a single variable.
b. Solve radical equations of a single variable, including those with extraneous roots.
c. Solve absolute value and compound inequalities of a single variable.
d. Add, subtract, multiply, and divide rational expressions and solve rational equations.
e. Simplify algebraic expressions involving negative and rational exponents.
Objective 2: Solve systems of equations and inequalities.
a. Solve systems of linear, absolute value, and quadratic equations algebraically and graphically.
b. Graph the solutions of systems of linear, absolute value, and quadratic inequalities on the coordinate plane.
c. Solve application problems involving systems of equations and inequalities.
Objective 3: Represent and compute fluently with complex numbers.
a. Simplify numerical expressions, including those with rational exponents.
b. Simplify expressions involving complex numbers and express them in standard form, a + bi.
Objective 4: Model and solve quadratic equations and inequalities.
a. Model real-world situations using quadratic equations.
b. Approximate the real solutions of quadratic equations graphically.
c. Solve quadratic equations of a single variable over the set of complex numbers by factoring, completing the square, and using the quadratic formula.
d. Solve quadratic inequalities of a single variable.
e. Write a quadratic equation when given the solutions of the equation.
Standard II: Students will understand and represent functions and analyze function behavior.
Objective 1: Represent mathematical situations using relations.
a. Model real-world relationships with functions.
b. Describe a pattern using function notation.
c. Determine when a relation is a function.
d. Determine the domain and range of relations.
Objective 2: Evaluate and analyze functions.
a. Find the value of a function at a given point.
b. Compose functions when possible.
c. Add, subtract, multiply, and divide functions.
d. Determine whether or not a function has an inverse, and find the inverse when it exists.
e. Identify the domain and range of a function resulting from the combination or composition of functions.
Objective 3: Define and graph exponential functions and use them to model problems in mathematical and real-world contexts.
a. Define exponential functions as functions of the form, b > 0, b ≠ 1.
b. Model problems of growth and decay using exponential functions.
c. Graph exponential functions.
Objective 4: Define and graph logarithmic functions and use them to solve problems in mathematics and real-world contexts.
a. Relate logarithmic and exponential functions.
b. Simplify logarithmic expressions.
c. Convert logarithms between bases.
d. Solve exponential and logarithmic equations.
e. Graph logarithmic functions.
f. Solve problems involving growth and decay.
Standard III: Students will use algebraic, spatial, and logical reasoning to solve geometry and measurement problems.
Objective 1: Examine the behavior of functions using coordinate geometry.
a. Identify the domain and range of the absolute value, quadratic, radical, sine, and cosine functions.
b. Graph the absolute value, quadratic, radical, sine, and cosine functions.
c. Graph functions using transformations of parent functions.
d. Write an equation of a parabola in the form y = a(x − h)2 + k when given a graph or an equation.
Objective 2: Determine radian and degree measures for angles.
a. Convert angle measurements between radians and degrees.
b. Find angle measures in degrees and radians using inverse trigonometric functions, including exact values for special triangles.
Objective 3: Determine trigonometric measurements using appropriate techniques, tools, and formulas.
a. Define the sine, cosine, and tangent functions using the unit circle.
b. Determine the exact values of the sine, cosine, and tangent functions for the special angles of the unit circle using reference angles.
c. Find the length of an arc using radian measure.
d. Find the area of a sector in a circle using radian measure.
Standard IV: Students will understand concepts from probability and statistics and apply statistical methods to solve problems.
Objective 1: Apply basic concepts of probability.
a. Distinguish between permutations and combinations and identify situations in which each is appropriate.
b. Calculate probabilities using permutations and combinations to count events.
c. Compute conditional and unconditional probabilities in various ways, including by definitions, the general multiplication rule, and probability trees.
d. Define simple discrete random variables.
Objective 2: Use percentiles and measures of variability to analyze data.
a. Compute different measures of spread, including the range, standard deviation, and interquartile range.
b. Compare the effectiveness of different measures of spread, including the range, standard deviation, and interquartile range in specific situations.
c. Use percentiles to summarize the distribution of a numerical variable.
d. Use histograms to obtain percentiles.
Utah State Standards
4.1.a Distinguish between permutations and combinations and identify situations in which each is appropriate
4.1.b Calculate probabilities using permutations and combinations to count events
ACT Warm up:
**Exploration – Permutations and Probability
Outcome-
Ex. Possible outcomes to flipping a coin.
Sample space-
Event-
Independent events-
Ex.
Dependent events-
Ex.
Fundamental Counting Principle
Ex for Independent Events. A sandwich menu offers customers a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayonnaise. How many different combinations of bread and spread are possible?
Ex for more than 2 Independent Events. Many voice mail systems allow owners to call from another phone and get their messages by entering a 3-digit code. How many codes are possible?
Ex. for Dependent Events. Carla wants to take 6 different classes next year. Assuming that each class if offered each period, how many different schedules could she have?
Factorials-
Notation
On a calculator, look for the ! in the MATH button menu.
By definition, 0! = 1.
Permutation-
Ex.
Notation-
Formula for Permutations
Ex. Eight people entered the Best Pizza Contest. How many ways can blue, red, and green ribbons be awarded?
Formula for Permutations with Repetitions.
Ex. How many different ways can the letters of the word BANANA be rearranged?
Ex. How many different ways can the letters of the word MISSISSIPPI be rearranged?
Combination-
Ex.
Notation-
Formula for Combination
Ex. Five cousins at a family reunion decide that three of them will go to pick up a pizza. How many ways can they choose the three people to go?
Ex. Six cards are drawn from a standard deck of cards. How many hands consist of two hearts and four spades?
**Exploration – Winning the Lottery
Homework Problems. p. 635 #11-19 odds
Homework Problems. p. 641 #13-19 odd, 23-29 odd
Utah State Standards
4.1.d Define simple discrete random variables
4.2.c Use percentiles to summarize the distribution of a numerical variable
4.2.d Use histograms to obtain percentiles
ACT Warm up:
Probability-
Notation
(different than permutation notation)
*you need to calculate the possible number of outcomes!
(sometimes this involves the Fundamental Counting Principle)
Practice Problems.
1. When two coins are tossed, what is the probability that both are tails?
2. When three coins are tossed, what is the probability that all three are heads?
3. Roman has a collection of 26 books- 16 are fiction and 10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability that he chooses 4 fiction and 4 nonfiction?
Random variable-
Ex. D represents the number showing when rolling a die.
Possible values:
Notation
P(X=n)
Probability distribution-
Ex. Probability distribution for rolling a die.
Histogram-
Ex. Probability distribution of the sum of the numbers of two dice rolled.
S=Sum / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12Probability / 1/36 / 1/18 / 1/12 / 1/9 / 5/36 / 1/6 / 5/36 / 1/9 / 1/12 / 1/18 / 1/36
Draw a histogram.
Using the histogram, which outcome is most likely? What is the probability?
Use the table and histogram to find P(S=9). What other sum has the same probability?
Homework Problems. p. 648 #19-29 all, 57-60.
Utah State Standards
4.1.c Compute conditional and unconditional probabilities in various ways, including by definitions and probability trees
ACT Warm up:
Probability of Independent Events
If two events, A and B are independent, then the probability of both events occurring P(A and B) is ______.
*This formula can be applied to any number of independent events.
Ex. Gerardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and replaces it. He then randomly selects another coin. What is the probability that both of the coins he selects are dimes?
Ex. When three dice are rolled, what is the probability that two dice show a 5 and the third die shows an even number?
Probability of Dependent Events
If two events, A and B, are dependent, then the probability of both events occurring P(A and B) is ______.
*This formula can be applied to any number of dependent events.
Ex. The host of a game show is drawing chips from a bag to determine the prizes for which contestants will play. The host draws from a bag of 20 chips, of which 11 say computer, 8 say trip, and 1 says truck. Drawing at random and without replacement, find each of the following probabilities.
a. What is the probability that the host draws a trip, then a computer.
b. What is the probability that the host draws a truck, then two trips.
Homework Problems. p.655 #15-23 odd, 31, 33
CDAS: Probability
Utah State Standards
4.2.a Compute and compare different measures of spread, including the range, standard deviation, and interquartile range
4.2.b Compare the effectiveness of different measures of spread, including the range, standard deviation, and interquartile range in specific situations
ACT Warm up:
Measures of Central Tendency
Mean-
Use when the data are spread out and you want an average of the values.
Median-
Use when the data contain outliers or points that are extremely high or low.
Mode-
Use when the data are tightly clustered around one or two values.
Ex. A new Internet company has 3 employees who are paid $300,000, 10 who are paid $100,000, and 60 who are paid $50,000.
1. Find the mean, median and mode.
2.Which measure of central tendency best represents the pay at this company?
3. Which measure of central tendency would recruiters for this company be most likely to use to attract job applicants?
Measures of Spread, Variation, or Dispersion
Range-
*Only gives information of the high and
the low values including outliers!
*Does not give information on the
majority of values
Variance-
Notation
Steps to Calculate Variance
1. Find the mean
2. Find the difference between each value in the set of data and the mean.
3. Square each difference.
4. Find the mean of the squares.
Standard Deviation-
*The typical variation for the data items from the mean.
Ex. The following table shows the length in thousands of miles of some of the longest rivers in the world. Find the standard deviation for these data.
River / Length(thousands of miles)
Nile / 4.16
Amazon / 4.08
Missouri / 2.35
Rio Grande / 1.90
Danube / 1.78
Interquartile Range-
*Should be used when data set contains outliers
Steps to Calculate
1. Find the median of the data set.
2. Find the median of the 1st half of the data Q1 (lower quartile)
3. Find the median of the 2nd half of the data Q3 (upper quartile)
4. Find the difference between Q3 and Q1
Ex.A year ago, Angela began working at a computer store. Her supervisor asked her to keep a record of the number of sales she made each month.
The following data set is a list of her sales for the last 12 months:
34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37
Use Angela's sales records to find:
a) the median
b) the range
c) the upper and lower quartiles
d) the interquartile range
Box Plots:
Draw a box plot for Angela’s sales records above.
**Exploration – Pulse Rates
Homework Problems. p. 667 #11, 13, 17,18, 24-26, worksheet on interquartile range
CDAS: Percentiles and Variability
Utah State Standard I: Students will use the language and operations of algebra to evaluate, analyze, and solve problems
ACT Warm up:
Properties of Real Numbers
Identity Properties:
Identity Property of Addition:
Ex: can be written as . (I know, it seems silly, but wait!)
Ex: Why is the same as ?
Identity Property of Multiplication:
Ex. Simplify: (Show two ways)
Ex. Simplify:
Inverse Properties:
Inverse Property of Addition:
Ex. What are the additive inverses of the following numbers?
a. 4 b. c. – 5
Inverse Property of Multiplication:
are called multiplicative inverses or reciprocals of each other.
Ex. What are the multiplicative inverses of the following numbers?
a. 10 b. c.
Properties of Equality
Commutative Property of Addition: For any real numbers
Ex. 4 + 3 =
Commutative Property of Multiplication: For any real numbers
Ex. The array shows a representation of the product .
How might this array also represent the product ?
Explain why is there no Commutative Property of Subtraction or Commutative Property of
Division? Illustrate with specific examples.