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Honors Algebra 2 Spring 2012 Ms. Katz.

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1 Honors Algebra 2 Spring 2012 Ms. Katz

2 Day 1: January 30th Objective: Form and meet study teams. Then work together to share mathematical ideas and to justify strategies as you represent geometric objects and order a series of connected functions to create a desired output. Seats Problems 1-1 to 1-2 Introduction: Ms. Katz, Books, Syllabus, Index Card Homework Record, Expectations Conclusion Homework: Have Parent/Guardian fill out last page of syllabus and sign; Problems 1-4 to 1-9 AND 1-13 to 1-19; Extra credit tissues or hand sanitizer (1) 2

3 Five Point Star

4 Function Notation The f is the name of the function machine, and the expression to the right of the equal sign shows what the machine does to any input.

5 Which do you prefer to write?
Function Notation The f is the name of the function machine and the value inside the parentheses is the input. The expression to the right of the equal sign shows what the machine does to the input. 25 Which do you prefer to write? Evaluate f when x = 25? OR 5

6 Function Machines (a) 6 -16 7 121 11

7 Function Machines (b) 64 8 -36 17 131065

8 Support www.cpm.org www.hotmath.com My Webpage on the HHS website
Resources (including worksheets from class) Extra support/practice Parent Guide Homework Help All the problems from the book Homework help and answers My Webpage on the HHS website Classwork and Homework Assignments Worksheets Extra Resources

9 Respond on Index Card: When did you take Algebra 1? Geometry?
Who was your Algebra 1 teacher? Geometry teacher? What grade do you think you earned in Geometry? What is one concept/topic from Algebra 1 that Ms. Katz could help you learn better? What grade would you like to earn in Algebra 2? (Be realistic) What sports/clubs are you involved in this Spring? My address (for teacher purposes only) is:

10 Day 2: January 31st Objective: Review expectations for class and homework. Work together to share mathematical ideas and to justify strategies as you order a series of connected functions to create a desired output. THEN Draw complete graphs of functions and identify possible inputs, outputs, and key points for describing those graphs. You will use a graphing calculator and develop presentation skills. HW Check and Correct (in red) Problems 1-10 and 1-12 Problem 1-27 Conclusion Homework: Problems 1-20 to 1-26; GET SUPPLIES; Extra credit tissues or hand sanitizer (1) 10

11 Complete Graph When a problem says graph an equation or draw a graph:
On graph paper: Plot key points accurately (-2,0) (3,0) Scale your axes appropriately x (0,-6) (.5,-6.25) Label the axes (with units if appropriate)

12 Day 3: February 1st Objective: Identify the domain and range of functions while improving your graphing-calculator skills. THEN Find points of intersection using multiple representations and learn how to use the [CALC], [TABLE], and [TBLSET] functions on a graphing calculator. HW Check and Correct (in red) Problems 1-28 to 1-34 Notes Problems 1-42 and 1-46 Conclusion Homework: Problems 1-35 to 1-41 AND 1-47 to 1-53; GET SUPPLIES; Extra credit tissues or hand sanitizer 12

13 Vertical Line Test If a vertical line intersects a curve more than once, it is not a function. Use the vertical line test to decide which graphs are functions.

14 Vertical Line Test If a vertical line intersects a curve more than once, it is not a function. Use the vertical line test to decide which graphs are functions.

15 Complete Graph When a problem says graph an equation or draw a graph:
On graph paper: Plot key points accurately (-2,0) (3,0) Scale your axes appropriately x (0,-6) (.5,-6.25) Label the axes (with units if appropriate)

16 The domain and range help determine the window of a graph.
Definitions Domain All possible input values (usually x), which allow the function to work. Range All possible output values (usually y), which result from using the function. The domain and range help determine the window of a graph.

17 1-34: Learning Log Title: Domain and Range
Describe everything you know about domain and range. Why are the domain and range important when graphing? What calculator buttons allow us to see the appropriate domain and range of a graph? 17

18 Symbols for Number Set Natural Numbers:
Counting numbers (maybe 0, 1, 2, 3, 4, and so on) Integers: Positive and negative counting numbers (-2, -1, 0, 1, 2, and so on) Rational Numbers: a number that can be expressed as an integer fraction (-3/2, -1/3, 0, 1, 55/7, 22, and so on) a number that can NOT be expressed as an integer fraction (π, √2, and so on) Irrational Numbers: NONE

19 Symbols for Number Set Real Numbers:
The set of all rational and irrational numbers Rational Numbers Integers Irrational Numbers Real Number Venn Diagram: Natural Numbers

20 less than or equal (included) greater than or equal (included)
Inequality Notation < > Less than (not included) Greater than (not included) less than or equal (included) greater than or equal (included) Open Dot and Parentheses ( ) Closed Dot and Brackets [ ] 20

21 Example: Inequalities
Graphically and algebraically represent the following: All real numbers greater than 11 Graph: Symbolic: OR

22 Example: Inequalities
Describe and algebraically represent the following: Description: Symbolic: All real numbers less than or equal to -5 OR

23 Example: Inequalities
Describe and graphically represent the following: Description: Graph: OR All real numbers greater than or equal to 1 and less than 5

24 Example: Inequalities
Graphically and algebraically represent the following: All real numbers less than -2 or greater than 4 Graph: Symbolic: OR

25 Day 4: February 2nd Objective: Find points of intersection using multiple representations and learn how to use the [CALC], [TABLE], and [TBLSET] functions on a graphing calculator. THEN Investigate a function defined by a geometric relationship and generate multiple algebraic representations for the function. HW Check and Correct (in red) Wrap-Up Notes Problems 1-42 and 1-46 Problems 1-54 to 1-58 Conclusion Homework: Problems 1-60 to 1-71; Get Supplies! Team Test Tuesday (?) 25

26 less than or equal (included) greater than or equal (included)
Inequality Notation < > Less than (not included) Greater than (not included) less than or equal (included) greater than or equal (included) Open Dot and Parentheses ( ) Closed Dot and Brackets [ ] 26

27 Multiple Representations
Non-Algebraic Table Rule or Equation Graph Context Algebraic

28 Solving a System Algebraically
Use the equations to solve the following system:

29 Using a Table to Solve a System
Use tables to solve the following system: X Y -3 39 -2 24 -1 13 6 1 3 2 4 9 18 5 31 X Y -3 15 -2 24 -1 29 30 1 27 2 20 3 9 4 -6 5 -25

30 Day 5: February 3rd Objective: Investigate a function defined by a geometric relationship and generate multiple algebraic representations for the function. THEN Develop an understanding of what it means to investigate a function as the family of hyperbolas is investigated. HW Check and Correct (in red) Finish Problems 1-57 to 1-58 Problems 1-78 to 1-83 Start Problems 1-99 to 1-104 Conclusion – [Project will be assigned next week] Homework: Problems 1-72 to 1-77 AND 1-84 to 1-90; Supplies! Team Test Monday? Tuesday? 30

31 Domain and Range Domain: Domain: All ℝ Range: Range:

32 Day 6: February 6th Objective: Develop an understanding of what it means to investigate a function as the family of hyperbolas is investigated. THEN Identify what all linear functions have in common and determine whether relationships in tables and situations are linear. HW Check and Correct (in red) Finish Problems 1-78 to 1-83 Problems 1-99 to 1-104 Assign Project and Review Rubric Start notes on Exponents if time Conclusion Homework: Problems 1-91 to 1-98 AND to 1-111 Ch. 1 Team Test Tomorrow Ch. 1 Individual Test Friday 32

33 Function Investigation Questions
What is the domain of the function? What is the range? Does the function have symmetry? What are the important/key points of this function? Why are they important? What is the shape of the graph? Does the function have any “problem points” or asymptotes? Why do they happen?

34 Hyperbola What to address: Domain and Range
Key Points (max/min, intercepts, etc) Asymptotes (a line that the graph of a curve approaches) Symmetry x -6 -1 1 1.5 1.75 1.9 1.99 2 2.01 2.1 2.25 2.5 3 4 5 10 y -.125 -.33 -.5 -2 -4 -10 -100 Ǿ 100 .5 .33 .125

35 (Multiple Values/Vary)
Parameter vs. Variable Variable (Multiple Values/Vary) Parameter (Specific/Constant)

36 Day 7: February 7th Objective: Assess Chapter 1 in a team setting. THEN Identify what all linear functions have in common and determine whether relationships in tables and situations are linear. HW Check and Correct (in red) Chapter 1 Team Test (≤ 50 minutes) Finish Problems 1-99 to 1-104 Review Project Rubric Start notes on Exponents if time Conclusion Homework: Problems to AND CL1-120 to CL1-124 Ch. 1 Individual Test Friday 36

37 Day 8: February 8th Objective: Identify what all linear functions have in common and determine whether relationships in tables and situations are linear. THEN Explore, state, and practice the rules for simplifying exponential expressions. HW Check and Correct (in red) Finish Problems 1-99 to 1-104 Notes on Exponents Practice – “Exponent Mania” Conclusion Homework: Problems CL1-125 to AND Exponent Mania Ch. 1 Individual Test Friday Problem 1-112(b) Due Monday, February 13th 37

38 1-104: Learning Log Title: Recognizing Linear Relationships
How do you recognize a linear relationship without a graph? How can you recognize a linear equation? How do you recognize a linear table? How do you recognize linear situation? What must the rate of change be for every relationship? 38

39 Base raised to an exponent
Exponential Notation BaseExponent Base raised to an exponent

40 Goal To write simplified statements that contain distinct bases, one whole number in the numerator and one in the denominator, and no negative exponents. Ex:

41 Exploration Evaluate the following without a calculator: 34 = 33 =
32 = 31 = Describe a pattern and find the answer for: 30 = 81 27 9 3 1

42 Anything to the zero power is one
1 Anything to the zero power is one Stress same base Can “a” equal zero? No. You can’t divide by 0. 42

43 Exploration Simplify: =

44 Product of a Power If you multiply powers having the same base, add the exponents. Stress same base 44

45 Example Simplify: =

46 Exploration Simplify:

47 Power of a Power To find a power of a power, multiply the exponents.
Stress same base 47

48 Example Simplify: = 8s13t11

49 Exploration Simplify:

50 If a base has a product, raise each factor to the power
Power of a Product Stress same base If a base has a product, raise each factor to the power 50

51 Example Simplify: = -288x7y20

52 Complete the tables (with fractions) by finding the pattern.
Exploration 55 3125 54 625 53 125 52 25 51 5 50 1 5-1 5-2 5-3 5-4 1/32 1/16 1/8 1 Complete the tables (with fractions) by finding the pattern. 2 1/5 4 1/25 1/125 8 1/625 16

53 A Negative Exponent A simplified expression has no negative exponents.
Stress same base A simplified expression has no negative exponents. 53

54 Example Simplify:

55 Exploration Simplify: =

56 Quotient of a Power To find a quotient of a power, subtract the bottom exponent from the top if the bases are the same. Stress same base 56

57 Example Simplify:

58 Exploration Simplify:

59 Power of a Quotient Stress same base To find a power of a quotient, raise the denominator and numerator to the same power. 59

60 Example Simplify:

61 Day 9: February 9th Objective: Represent exponential growth with a diagram, table, equation, and graph. Write equations based on the patterns in tables, recognize patterns of exponential growth, and use equations to make predictions. HW Check and Correct (in red) Problems 2-1 to 2-5 Conclusion/Notes Homework: Problems 2-6 to 2-12 (finish thru 2-19 if you don’t want weekend HW aside from the project) Ch. 1 Individual Test TOMORROW Problem 1-112(b) Due Monday, February 13th 61

62 Multiplying Like Bunnies
Case 1: Start with 2 rabbits; each pair has 2 babies per month Month TOTAL Babies born this month 2 1 3 2 4 4 8 8 16

63 Multiplying Like Bunnies
Case 2: Start with 10 rabbits; each pair has 2 babies per month Month TOTAL Babies born this month 10 1 2 3 10 20 20 40 40 80

64 Multiplying Like Bunnies
Case 3: Start with 2 rabbits; each pair has 4 babies per month Month TOTAL Babies born this month 2 1 3 4 6 12 18 36 54

65 Multiplying Like Bunnies
Case 4: Start with 2 rabbits; each pair has 6 babies per month Month TOTAL Babies born this month 2 1 3 6 8 24 32 96 128

66 Day 10: February 10th Objective: Assess Chapter 1 in an individual setting. Silence your cell phone and put it in your school bag (not your pocket) Get a ruler, pencil/eraser, and calculator out First: Calculator Portion…put calculator away when finished Second: Non-Calculator Portion (ask for it) Third: Correct last night’s homework Homework: Finish your project! Problems 2-13 to 2-19 Individual Take Home: Problem 1-112(b) Due Monday! Make sure the rubric is already attached by 3rd block! 66 66

67 Day 11: February 13th Objective: Generate data and model the data with tables, rules, and graphs. Calculate the rebound ratio when a ball bounces. THEN Introduce an example of exponential decay. HW Check and Correct (in red) Hand in Project with Rubric Attached! Wrap-Up Lesson (LL Stuff) Problems 2-21 to 2-23 Problems 2-30 to 2-35 Conclusion Homework: Problems 2-24 to 2-29 AND 2-36 to 2-41 67

68 Exponential Table There is a constant multiplier between consecutive output values. X Y -1 5.67 17 1 51 2 153 3 459 x3 x3 x3 x3

69 Exponential Graph Notes: Horizontal Asymptote

70 Doubles Recursive Formula
A formula that requires the previous terms in order to find the value of the next term. Example: 2, 4, 8, 16, … Recursive Formula: Doubles

71 Plug in with Parentheses in the calculator!
Explicit Formula A formula that requires the number of the term in order to find the value of the next term. Example: 2, 4, 8, 16, … Explicit Formulas: Plug in with Parentheses in the calculator! Rate Initial Month

72 Exponential Equation Rate Initial
A function whose input (x) is located in the exponent. Example: Jason has $17 and quadruples his money every month. Write an equation to represent the situation. Rate Initial

73 2-5: Learning Log Title: Exponential Functions
Describe everything you know about exponential functions. What operation must be in the equation? What do their graphs look like? What patterns are in the tables? Draw examples of all the representations. 73 73 73

74 Rebound Ratio

75 Continuous Graph The points of the graph are connected. Therefore, there are no holes or breaks in it.

76 Discrete Graph The graph is made up of separate points.

77 Starting Height vs. Rebound Height
y = mx + b y = mx Rebound Height Rebound Ratio Starting Height Starting Height

78 Day 12: February 14th Objective: Introduce an example of exponential decay. THEN Introduce sequences and sort them into groups based on patterns in their representations. Also, identify sequences generated by adding a constant as arithmetic, and those generated by multiplying a constant as geometric. HW Check and Correct (in red) Problems 2-30 to 2-35 Problems 2-42, 2-43, and 2-45 Conclusion Homework: Problems 2-46 to 2-60 78

79 Reminders/Notes You are responsible for content in Math Notes boxes – make sure you review them. There is a non-calculator portion to every test. Work on your pacing. All tests are cumulative. Sometimes homework problems introduce topics that won’t be taught in class. You get out of this class what you put in – make sure you are doing your part. If you need help, please see me before it’s too late. As Mellor’s sign states, “TODAY is the day to worry about your grade.” 79 79 79

80 Bounce vs. Rebound Height
Bounce Number Rebound Height y = abx Rebound Height Starting Height Rebound Ratio Bounce Number Discrete!

81 Exponential Function Where r is the rebound ratio, n is the bounce number, and y is the height of the ball after the nth bounce.

82 Summary of Bounce Labs Lesson 2.1.2:
The height of a ball’s rebound grows constantly as the drop height grows, so it makes sense that this would be a linear model. Lesson 2.1.3: The height of each bounce is a constant multiple of its previous height, so it makes sense that, if left to bounce repeatedly, the ball’s height would shrink exponentially. 82 82 82

83 Day 13: February 15th Objective: Introduce sequences and sort them into groups based on patterns in their representations. Also, identify sequences generated by adding a constant as arithmetic, and those generated by multiplying a constant as geometric. THEN Learn the vocabulary and notation for arithmetic sequences as formulas for the nth term are developed. HW Check and Correct (in red) Finish Problems 2-43, and 2-45 Problems 2-61 to 2-70 Start Problem 2-78 if time Conclusion Homework: Problems 2-71 to 2-77 83

84 Sequences vs. Functions
Sequence: t(n) Function: f(x) Domain (n) = Positive Integers (sometimes 0) Range (t(n)) = Can be all Real numbers The Graph is Discrete Domain (x) = Can be all Real numbers Range (f(x))= Can be all Real numbers The Graph is Continuous

85 Arithmetic Sequences A sequence which has a constant difference between terms. The rule is linear. Example: 1, 4, 7, 10, 13,… (generator is +3) n t(n) 1 4 2 7 3 10 13 +3 Discrete +3 +3 +3

86 Geometric Sequences A sequence which has a constant ratio between terms. The rule is exponential. Example: 4, 8, 16, 32, 64, … (generator is x2) n t(n) 4 1 8 2 16 3 32 64 x2 Discrete x2 x2 x2

87 Day 14: February 16th Objective: Use geometric sequences to solve problems involving percent increase and decrease. Also, identify multipliers both to classify the sequences as geometric and to write equations for those sequences. THEN Recognize that sequences are functions with domains limited to non-negative integers. Use Guess and Check or graphical methods to solve exponential equations. HW Check and Correct (in red) Review Problems 2-62 to 2-68 Review Chapter 1 Individual Test (& general comments) Problems 2-78 to 2-82 Problems 2-92 to 2-97 Conclusion Homework: Problems 2-86 to 2-91 AND 2-98 to 2-105 87

88 Day 15: February 17th Objective: Use geometric sequences to solve problems involving percent increase and decrease. Also, identify multipliers both to classify the sequences as geometric and to write equations for those sequences. THEN Recognize that sequences are functions with domains limited to non-negative integers. Use Guess and Check or graphical methods to solve exponential equations. HW Check and Correct (in red) Problems 2-78 to 2-82 Problems 2-92 to 2-97 Conclusion Homework: Problems to AND to AND Revisit other homework problems that you may have had troubles with – slow down, redo and regroup for Monday. 88

89 Generator for a Percent Increase
What is a 15% increase of 100? First Step: Second Step: . . 15.% = . 0.15 2 1 Multiplier

90 Generator for a Percent Decrease
Example: What is the multiplier for a 17% decrease? First Step: Second Step: . . 17.% = . 0.17 2 1 Multiplier

91 πPod Problem Week Sales 1 2 3 4 n 100 = 115 = = 132.25 = = 152.09 = = 174.9 = = n

92 Day 16: February 21st Objective: Recognize that sequences are functions with domains limited to non-negative integers. Use Guess and Check or graphical methods to solve exponential equations. THEN Write rules for arithmetic and geometric sequences, identifying the first term as term number one rather than term zero. THEN Identify equivalent expressions and develop and share algebraic strategies for demonstrating equivalence. HW Check – Compare answers with teammates, please! Finish Problems 2-92 to 2-97 Review Problems to [Do 2-109] Problems to 2-120 Conclusion Homework: Problems to AND to 2-129 Chapter 2 Team Test Thursday 92

93 Exponential Function Time Initial Rate

94 A(t) = P ( 1 + r ) t Exponential Growth
A(t): Amount as a function in terms of t P: Principal (starting amount) t: Time after starting point r: Decimal increase (% ÷ 100) Learning Log

95 A(t) = P ( 1 – r ) t Exponential Decay
A(t): Amount as a function in terms of t P: Principal (starting amount) t: Time after starting point r: Decimal decrease (% ÷ 100) Learning Log

96 2-94: Learning Log Title: Sequences vs. Functions
Is a sequence a function? What makes a sequence different than most functions? If a sequence and a function have the same rule, how are they different? What are the restrictions on the domain of a sequence? 96 96 96

97 Working Backwards for a Rule
First find the generator and the n=0 term. Then write the equation: Ex 1: Ex 2: 36, 32, 28, 24, … 40, – 4 Sequences start with n=1 now! 3, -15, 75, -375, … x-5

98 Day 17: February 22nd Objective: Identify equivalent expressions and develop and share algebraic strategies for demonstrating equivalence. THEN Use an area model to multiply expressions. Factor expressions and demonstrate equivalence. THEN Solve equations by first rewriting them as simpler equivalent equations. HW Check and Correct (in red) Finish Problems to 2-120 Review Problems to 2-134 Problems to 2-147 Hand Back and Review Projects Conclusion Homework: Problems to AND to 2-156 Chapter 2 Team Test Tomorrow 98

99 Properties Distributive Property a(b + c) = ab + ac
(a + b)(c + d) = ac + ad + bc + bd Associative Property Addition: a + (b + c) = (a + b) + c Multiplication: a(bc) = (ab)c Commutative Property Addition: a + b = b + a Multiplication: ab = ba

100 Two Butt Cheeks When there is addition or subtraction:

101 Day 18: February 23rd Objective: Solve equations by first rewriting them as simpler equivalent equations. THEN Assess Chapter 2 in a team setting. HW Check and Correct (in red) Start Problems to 2-147 Chapter 2 Team Test Conclusion Homework: Problems CL2-157 to CL2-165 Chapter 2 Individual Test Tuesday 101

102 Day 19: February 24th Objective: Solve equations by first rewriting them as simpler equivalent equations. THEN Investigate the family of functions y = bx. Make and justify statements about the behaviors of graphs in this family. HW Check and Correct (in red) Work a little on Problems to 2-147 Problems 3-1 to 3-6 Conclusion Homework: Problems 3-7 to 3-21 Chapter 2 Individual Test Tuesday 102

103 Solutions to Equations in 2-144
x = -5, 4 x = -2, ½ x = 1 x = 2 x = 2 and y = -5 x = -5 and y = 3

104 Begin to Investigate Exponentials
If the graph shows nothing, try: ZOOM – 4:ZDecimal 3-2 needs SKETCHES OF GRAPHS and SUMMARY STATEMENTS

105 3-2: Possible Exponential Graphs
Answer the following questions: What is the shape of the graph? (Sketch) What are the domain and range? What are the intercepts? Or other key points? Are there any points that don’t work? Or asymptotes?

106 X b < 0 Graphs of y=bx x y=(-2)x -3 -2 -1 1 2 3 4 -.125 .25 -.5 1
1 2 3 4 -.125 .25 -.5 1 -2 4 -8 16

107 b > 0 Graphs of y=bx b = 0 0 < b < 1 b = 1 b > 1
x-axis where x>0 Decreasing, asymptote y=0 No x-intercepts. The y-intercept is (0,1). There is never a vertical asymptote b = 1 b > 1 Horizontal line y=1 Increasing, asymptote y=0

108 Day 20: February 27th Objective: Deepen and extend our knowledge of exponential functions by examining the relationships between different representations of those functions. Generalize the roles of a and b in y = a·bx. THEN Apply our knowledge of linear and exponential functions to investigate the relationship between simple and compound interest. ***NEW SEATS*** HW Check and Correct (in red) Wrap-Up Problem 3-6 Review Chapter 2 Team Test Problems 3-22, 23, 25 Start Problems 3-34 to 3-38 Exponential Graph Online Homework: Problems 3-26 to 3-33 AND STUDY! Chapter 2 Individual Test Tomorrow 108

109 3-6: Learning Log Title: Investigating y = bx
What values of b are acceptable? What values of b are unacceptable? Why? How does changing the value of b affect a graph? What does the graph look like when 0<b<1? What does the graph look like when b=1? What does the graph look like when b>1? What is the horizontal asymptote? Why is there no vertical asymptote? 109 109 109

110 b > 0 Graphs of y=bx 0 < b < 1 b = 1 b > 1 Domain:
x-intercepts: y-intercept: Vertical asymptote: Decreasing, asymptote y=0 None Range: (0,1) None For any Exponential: b = 1 b > 0 b > 1 Horizontal line y=1 Increasing, asymptote y=0 Range: Range:

111 Exponential Function Web
Table Rule or Equation Graph Context

112 Graphs of y=abx a x b (0,a) a is the starting point Context:
The initial value The multiplier x y=a(b)x -3 -2 -1 1 2 x b (0,a) a a is the starting point Context: b is the multiplier

113 Day 21: February 28th Objective: Assess Chapter 2 in an individual setting. Silence your cell phone and put it in your school bag (not your pocket) Get a ruler, pencil/eraser, and calculator out First: Calculator Portion…put calculator away when finished Second: Non-Calculator Portion (ask for it) Third: Check your work & hand the test in to Ms. Katz Fourth: Correct last night’s homework & start tonight’s Homework: Problems 3-39 to 3-47 113 113

114 Day 22: February 29th Objective: Apply our knowledge of linear and exponential functions to investigate the relationship between simple and compound interest. THEN Represent exponential decay in multiple ways and further investigate the effect when the exponent is 0 or negative. HW Check and Correct (in red) Discuss 3-29 to 3-31 and 3-40 Problems 3-34 to 3-38 Exponential Graph Online Problems 3-48 to 3-52 Homework: Problems 3-53 to 3-61 114

115 3-40: Tickets for a Concert
w P(w) -3 -2 -1 150 1 162 2 3 4 5 6 7 8 119.07 ÷1.08 128.60 ÷1.08 138.89 ÷1.08 x1.08 x1.08 174.96 x1.08 Don’t change the value of b in the equation to calculate past outputs. Use negative inputs: 188.96 x1.08 204.07 x1.08 220.40 x1.08 238.03 x1.08 257.07 x1.08 277.04

116 Interest Example Find an equation for the following context: Fred invests $12,000 in an account that offers 3.2% annual interest compounded annually. . . . 3.2% = 0.032 2 1 Decimal Rate The initial value

117 Interest Example Find an equation for the following context: Fred invests $12,000 in an account that offers 3.2% annual interest compounded semiannually. . . 3.2% = 0.032 2 1 Decimal Rate The initial value Number of Intervals

118 Day 23: March 1st Objective: Represent exponential decay in multiple ways and further investigate the effect when the exponent is 0 or negative. HW Check and Correct (in red) Wrap-Up Problems 3-34 to 3-38 Problems 3-48 to 3-52 Homework: Problems 3-62 AND 3-64 to 3-71 118

119 Simple: Compound: Interest P = Principal Amount (original)
r = rate ( % ÷ 100 ) t = time in years n = number of intervals

120 Day 24: March 5th Objective: Represent exponential decay in multiple ways and further investigate the effect when the exponent is 0 or negative. THEN Use what is known about exponential growth to write equations for exponential functions presented as graphs. THEN Complete the exponential multiple-representations web, solidifying connections between the table, equation, graph, and context representations of an exponential function. HW Check and Correct (in red) Review Chapter 2 Individual Test Problems 3-48 to 3-52 Review Problems 3-62 to 3-63 Problems 3-72 to 3-77 Homework: Problems 3-78 to 3-86 AND 3-87 to 3-88 Chapter 3 Team Test Thursday 120

121 Penny Lab/Half-Life ÷0.5 ÷0.5 ÷0.5 x0.5 x0.5 x0.5 x0.5 x0.5 x0.5 x y
200 x y -3 -2 -1 1 2 3 4 5 6 800 ÷0.5 400 ÷0.5 200 ÷0.5 100 x0.5 100 50 x0.5 25 x0.5 12.5 x0.5 6.25 x0.5 3.125 x0.5 3 1.5625

122 3-63: Learning Log Title: Graph → Rule for Exponential Functions
Methods for creating an exponential rule given a graph: The y-intercept for “a,” and if you have consecutive terms divide the higher term by the lower term to find “b” Making a table and then use guess and check 122 122 122

123 Day 25: March 6th Objective: Complete the exponential multiple-representations web, solidifying connections between the table, equation, graph, and context representations of an exponential function. THEN Find equations of linear and exponential functions by using known quantities to solve for a missing parameter. Also, interpret fractional exponents. THEN Find linear and exponential equations given two points. Also, evaluate roots with the calculator by converting to fractional exponent notation. HW Check and Correct (in red) Continue Working on Problems 3-72 to 3-77 Review Problems 3-87 to 3-88 Problems 3-89 to 3-94 Problems to 3-108 Homework: Problems 3-95 to AND to 3-116 Chapter 3 Team Test Thursday 123

124 Exponential Function Web
Table Rule or Equation Graph Context

125 Example: 3-89 Find the equation of an exponential function with an asymptote at y = 0 that passes through the points (0,5) and (3,320).

126 Day 26: March 7th Objective: Find linear and exponential equations given two points. Also, evaluate roots with the calculator by converting to fractional exponent notation. THEN Write and solve a system of exponential functions in the context of investigating used-car prices. HW Check and Correct (in red) Wrap-Up Fractional Exponents (3 slides – LL) Problems to 3-108 Problems to 3-120 Homework: Problems to (and start closure?) Chapter 3 Team Test Tomorrow 126

127 Radical Property ONLY when a≥0 and b≥0

128 Exponents into Radical Notation
bp/q = Generally b≥0

129 Evaluate the following without a calculator:
Example Evaluate the following without a calculator:

130 System of Exponential Equations
Find an exponential function that passes through (2,16) and (6,256). Substitute into either equation to find a Substitute into y=abx twice Larger exponent first ÷ ( ) Subtract Exponents Divide #s Find the Root

131 Day 27: March 8th Objective: Assess Chapter 3 in a team setting. THEN Collect non-linear data, fit an equation to the data, and use the equation to make predictions. HW Check and Correct (in red) Quickly! Wrap-Up Problem 3-117 Chapter 3 Team Test Start Problems 4-1 to 4-4 Homework: Problems CL3-131 to AND 4-5 to 4-12 Chapter 3 Individual Test Thursday 131

132 By the End of the Chapter…
You will be able to easily sketch graphs similar to the following by just looking at the equations:

133 Shrinking Targets Lab Radius (mm) Weight (grams) 78 3.5 71 3.0 61 2.2 55 1.7 46.5 1.3 34 0.7 27 0.4 22 0.3

134 Day 28: March 9th Objective: Collect non-linear data, fit an equation to the data, and use the equation to make predictions. THEN Connect transformations of parabolas with their equations in graphing form. HW Check and Correct (in red) Quickly! Problems 4-1 to 4-4 Problems 4-13 to 4-17 Homework: Problems 4-18 to 4-33 Chapter 3 Individual Test Thursday Finish Closure (CL3-138 to 3-142) while you study for Thursday’s test [won’t be checked for points] 134

135 Modeling our Data Which measurement of a circle directly affects its weight? Area What is the equation for the area of a circle? What is the name of the graph of the area of a circle? Quadratic/Parabola

136 What we know about Transforming y=x2
y=ax2 The further the number you multiply by is from zero, the steeper the parabola. The closer the number you multiply by is to zero, the wider the parabola.

137 Vertical Dilations Transform! Vertical Stretch Vertical Compression

138 Day 29: March 12th Objective: Connect transformations of parabolas with their equations in graphing form. HW Check and Correct (in red) Quickly! Problems 4-13 to 4-17 Conclusion Homework: Problems CL3-138 to 3-142  I changed my mind…get points tomorrow Chapter 3 Individual Test Thursday 138

139 Day 30: March 13th Objective: Graph quadratic equations without making tables. Also, rewrite quadratic equations from standard form into graphing form. HW Check Quick Team Tests Wrap-Up Problems 4-16 to 4-17 Start Problems 4-34 to 4-38 Conclusion Homework: Problems 4-39 to 4-45 Chapter 3 Individual Test Thursday 139

140 Graphing Form for a Parabola
y = a(x – h)2 + k ( h, k ): The Vertex The value of a same opposite Positive: Opens Up If it Increases: Vertical Stretch Negative: Opens Down If it Decreases: Vertical Compression

141 –3 – 5 2 Use the stretch factor to translate the “first points”
Example Plot : y = 2(x+3)2 – 5 –3 2 – 5 Use the stretch factor to translate the “first points” Find the vertex Draw the Parabola Find the stretch factor

142 Day 31: March 14th Objective: Graph quadratic equations without making tables. Also, rewrite quadratic equations from standard form into graphing form. HW Check and Correct (in red) Quickly! Problems 4-35 to 4-38 Conclusion Homework: Study like it’s your job! Chapter 3 Individual Test Tomorrow 142

143 Graphing Form to Standard Form
Same a!

144 Day 32: March 15th Objective: Assess Chapter 3 in an individual setting. Silence your cell phone and put it in your school bag (not your pocket) Get a ruler, pencil/eraser, and calculator out First: Calculator Portion…put calculator away when finished Second: Non-Calculator Portion (ask for it) Third: Check your work & hand the test in to Ms. Katz Fourth: Correct last night’s homework Fifth: Work on Problems 4-37 and 4-38 and then re-do 4-40 Homework: Problems 4-37 and 4-38 AND 4-52 to 4-58 144 144

145 Day 33: March 16th Objective: Learn how to write quadratic equations for situations using the graphing form of the parabola y = a(x – h)2 + k. Specifically, develop an algebraic strategy for finding the value of the stretch factor, a. THEN Transform the graphs of y = bx, y = 1/x, y = √x, and y = x3. HW Check and Correct (in red) Quickly! Review Problems 4-37 to 4-38 Problems 4-46 to 4-50 Start Problems 4-59 to 4-63 Conclusion Homework: Problems 4-51 AND 4-64 to 4-70 145

146 Standard Form to Graphing Form
Use an algebraic method to write in graphing form. 1. Find the value of a: 2 3. Average the x-intercepts for h 2. Find the x-intercepts 4. Substitute h into the rule for k 5. Substitute a, h, k into the graphing form WARNING: This method does not work if there are no x-intercepts

147 Standard Form to Graphing Form
Use an algebraic method to write in graphing form. 1. Find the value of a: 2 3. Average the x-intercepts for h 2. Find the x-intercepts 4. Substitute h into the rule for k 5. Substitute a, h, k into the graphing form WARNING: This method does not work if there are no x-intercepts

148 Finding the Equation of a Line
Find the equation of a line that passes through the point (3,5) and has a slope of 2.

149 Finding a Quadratic Equation with the Vertex and Another Point
A rabbit jumped over a 3ft-high fence. The highest point the rabbit reached was 3 feet and it landed 8 feet from where it jumped. Assume the rabbit follows a parabolic path. Sketch a graph and find the equation for the height of the rabbit verses the horizontal distance it has traveled. Since we know the vertex Sketch: One Possibility: Substitute into y=a(x-h)2+k: (4,3) Plug in the vertex Plug in another point Label all the known points Solve for a (0,0) (8,0) Plug in a,h,&k Equation:

150 Different Forms For a Quadratic
Parent Graph: y = x2 Factored Form: y = __( __ )( __ ) Standard form: y = ax2 + bx + c Graphing Form: y = a(x – h)2 + k Vertex (locator point): ( h, k) Vertical Compression: Open up: Vertical Stretch: Open down: The “a” is the same in standard and graphing form!

151 Day 34: March 19th Objective: Transform the graphs of y = bx, y = 1/x, y = √x, and y = x3. THEN Identify the point (h,k) for parabolas, hyperbolas, cubics, and square root graphs, and relate the Point-Slope form of a line to (h,k). Consolidate all of the understanding of parent graphs and general equations in a toolkit. THEN Use our knowledge of transformations to write a general equation for a family of functions based on an absolute value parent graph. HW Check and Correct (in red) Quickly! Finish Problems 4-59 to 4-63 PPT Examples on Transformations Graphing Form Packet (Up through Linear) Problems 4-99 to 4-101 Conclusion Homework: Problems 4-71 to 4-85 151

152 Function Transformations
Family Parent Graphing Calculator Cubic Hyperbola Square Root Exponential

153 Parent Graph: When a=1, h=0, and k=0
Graphing Form ( h, k ): The Key Point The value of a Positive: Same Orientation If it Increases: Vertical Stretch Negative: Flipped If it Decreases: Vertical Compression Parent Graph: When a=1, h=0, and k=0 Quadratic Cubic Hyperbola Square Root Exponential Linear

154 Example: Quadratic Transformation: Shift the parent graph three units to the right and four units up. y = 4 New Equation: (3,4) x = 3

155 Transformation: Flip the parent graph and shift it five units up.
Example: Cubic Transformation: Flip the parent graph and shift it five units up. Transformation: y = 5 New Equation: (0,5) x = 0

156 Example: Hyperbola Transformation: Shift the parent graph four units to the left and three units down. Transformation: New Equation: (-4,3) y = -3 x = -4

157 Transformation: Shift the parent graph six units to the left.
Example: Square Root Transformation: Shift the parent graph six units to the left. Transformation: x = -6 New Equation: y = 0 (-6,0)

158 Example: Exponential Transformation:
Transformation: Shift the parent graph five units to the right and two units up. Then stretch the graph by a factor of 3. New Equation: a = 3 y = 2 (5,2) x = 5

159 Unless specified, you do not need to have the answer in y=mx+b form!
Linear Function Parent Equation Graphing Form (h,k) Point: Slope: Unless specified, you do not need to have the answer in y=mx+b form!

160 Example: Linear Transformation: A line with slope ½ that passes through the point (-6,4). Slope = ½ New Equation: (-6,4) y = 4 Point-Slope Form x = -6 Slope Point

161 Day 35: March 20th Objective: Use our knowledge of transformations to write a general equation for a family of functions based on an absolute value parent graph. THEN Use what we know about transforming parabolas to make conjectures about transforming relations, specifically sleeping parabolas and circles. Also, define the meaning of a non-function (relation). HW Check and Correct (in red) Quickly! Finish Problems 4-99 to & Add to Graphing Forms Packet Problems to 4-117 Start Problems to 4-134 Conclusion Homework: Problems 4-91 to 4-98 AND to 4-111 Ch. 4 Team Test Thursday Ch. 4 Individual Test next week 161

162 Go to the right once to NUM Choose 1:abs(
Absolute Value in a TI Hit MATH Go to the right once to NUM Choose 1:abs(

163 Absolute Value Function
Parent Equation Graphing Form MATH Right to NUM 1. abs( Absolute value can be found in the calculator:

164 Example: Absolute Value
Transformation: Flip the parent graph and shift it three units to the left and four units up. Transformation: (-3,4) y = 4 New Equation: x = -3

165 Sleeping Parabola Parent: Graphing Form: Calculator:

166 Circle Parent: Graphing Form: Calculator:

167 Day 36: March 21st Objective: Learn how to convert a parabola into graphing form by completing the square. THEN Extend the idea of completing the square to change circles written in standard form into graphing form. HW Check and Correct (in red) Quickly! Bell Ringer! (Next slide) Review Ch. 3 Individual Test Finish Problems to 4-117 Problems to 4-134 Start Problems to 4-146 Conclusion Homework: Problems to AND to 4-143 Ch. 4 Team Test Tomorrow [NO CALCULATORS!] Ch. 4 Individual Test next week 167

168 Distribute the following:
Bell Ringer Distribute the following: y = (x – 2)2 y = (x + 3)2 y = (x + 4)2 y = (x – 6)2 Factor the following: y = x2 + 8x + 16 y = x2 – 16x + 64 y = x2 + 20x + 100 y = x2 – 9x Is there a pattern when comparing a, b, and c when it is in standard form vs factored form?

169 (0,0) (h,k) Equation for a Circle Example Graphing Form
Center: Radius: Graphing Form (h,k) Center: Radius:

170 Transformation: A circle centered at (4,-1) whose radius is 4.
Example: Circle Transformation: A circle centered at (4,-1) whose radius is 4. Transformation: x = 4 New Equation: y = -1 (4,-1) (4,-1) Center: Radius: Is a circle a function? NO!

171 Day 37: March 22nd Objective: Assess Chapter 4 in a team setting. THEN Learn how to convert a parabola into graphing form by completing the square. THEN Extend the idea of completing the square to change circles written in standard form into graphing form. HW Check and Correct (in red) Quickly! Chapter 4 Team Test – No Calculators Finish Problems to 4-134 Start Problems to 4-146 Conclusion Homework: Problems to 4-155 Ch. 4 Individual Test Next Friday 171

172 Day 38: March 23rd Objective: Extend the idea of completing the square to change circles written in standard form into graphing form. HW Check and Correct (in red) Quickly! Summarize Problems to 4-134 Problems to 4-146 Start Lesson 5.1.1 Conclusion Homework: Problems CL4-156 to CL4-166 Ch. 4 Individual Test Next Friday – Start Studying! 172

173 Perfect Square A polynomial that can be factored into the following form: (x + a)2

174 Distribute the following:
Bell Ringer Distribute the following: y = (x – 2)2 y = (x + 3)2 y = (x + 4)2 y = (x – 6)2 Factor the following: y = x2 + 8x + 16 y = x2 – 16x + 64 y = x2 + 20x + 100 y = x2 – 9x Is there a pattern when comparing a, b, and c when it is in standard form vs factored form?

175 Completing the Square x2 + bx + c is a perfect square if:
The value of c will always be positive. Always write out all of your work. It will help you soon.

176 Completing the Square Find the c that completes the square:
x2 + 50x + c x2 – 22x + c x2 + 15x + c

177 Factoring a Completed Square
If x2 + bx + c is a perfect square, then it will easily factor to

178 Perfect Squares: Parabolas & Circles
Find the vertices of the following graphs and state whether they are maximums or minimums. y = (x + 5)2 – 5 y = -(x + 3)2 + 1 y = -3(x – 7)2 + 8 y = 4(x – 52)2 – 74 State the length of the radius and the coordinates of the center for each circle below: ( x – 2 )2 + ( y + 7 )2 = 64 x2 + y2 = 36 ( x + 4 )2 + ( y + 11 )2 = 5 ( x + 3 )2 + y2 = 175

179 A new Equation? What will the graph of the following look like:

180 Standard to Graphing: Quadratic
Find the vertex of the following equation by completing the square: y = x2 + 8x + 25 y = (x2 + 8x ) + 25 – 16 16 y = (x + 4)2 + 9 (-4, 9) Vertex:

181 Standard to Graphing: Quadratic
Find the vertex of the following equation by completing the square: y = 3x2 – 18x – 10 y = 3(x2 – 6x ) – 10 – 3 9 9 y = (x – 3)2 – 10 – 27 y = 3(x – 3)2 – 37 (3, -37) Vertex:

182 Standard to Graphing: Circle
Find the center and radius of the equation by completing the square: x2 + y2 + 6x – 12y – 9 = 0 x2 + 6x + y2 – 12y – 9 = 0 x2 + 6x + y2 – 12y = 9 (x2 + 6x ) + (y2 – 12y ) = 9 36 9 36 (x + 3)2 + (y – 6)2 = 54 (-3, 6) Center: Radius:

183 Solving Graphically How did you use the graph to solve:
(x+3)2 – 5 = 4? What other equations could you solve? x = -6 x = 0

184 Day 39: March 26th Objective: Solve a variety of equations and discuss different methods for solving them. Also, justify strategies and develop methods for checking solutions. THEN Use graphs to validate algebraic solutions and to approximate solutions when no algebraic method is available, and use two different methods to solve one-variable equations graphically. HW Check Quickly! Problems 5-3 to 5-5 Problems 5-13 to 5-17 Time? Start Lesson 5.1.2 Conclusion Homework: Problems 5-6 to 5-12 AND 5-18 to 5-24 Ch. 4 Individual Test Friday – Start Studying! [If you know you’re not going to be here due to extenuating circumstances, you must see me ahead of time to take the test.] 184

185 5-5: Learning Log Title: Strategies for Solving Equations
Summarize all of the solving strategies you saw today. Show an example of each strategy. Explain the type of equations for which each strategy works best. Make sure to explain Rewriting. Make sure to explain Undoing. Make sure to explain Looking Inside. 185 185 185

186 Extraneous Solutions

187 Calculator & Solving Equations
Method 1: Enter – Calculator Function – Method 2: Intersection CALC: intersect X-intercept x-coordinates! CALC: zero

188 Day 40: March 27th Homework: Problems 5-25 to 5-32 AND 5-37 to 5-43
Objective: Use graphs to validate algebraic solutions and to approximate solutions when no algebraic method is available, and use two different methods to solve one-variable equations graphically. THEN Solve systems of linear and non-linear equations using multiple strategies. Determine the number of solutions for systems and interpret solutions graphically. THEN Use problem solving to write equations and find solutions for real-life applications. HW Check and Correct (in red) Quickly! & Look at Ch. 4 Team Tests(?) Problems 5-16 to 5-17 Problems 5-33 to 5-36 Problems 5-44 to 5-47 Conclusion Homework: Problems 5-25 to 5-32 AND 5-37 to 5-43 Ch. 4 Individual Test Friday – Start Studying! [If you know you’re not going to be here due to extenuating circumstances, you must see me ahead of time to take the test.] 188

189 5-17: Learning Log Title: The Meaning of Solution, Part 1
What does the solution to an equation mean? Do you have any new ideas about solutions that you did not have before? Do you have any new methods to find solutions? How can you use the calculator to solve an equation? How can you use the intersection function on the calculator to find a solution? How can you use the zero function on the calculator to find a solution? Why are there equations that we can not solve algebraically yet? 189 189 189 189

190 5-34

191 Solve the Following Algebraically
OR

192 Solve the Following Algebraically

193 Day 41: March 28th Objective: Use problem solving to write equations and find solutions for real-life applications. THEN Extend what was learned about solving systems of equations graphically to solving systems of inequalities. HW Check and Correct (in red) Quickly! & Look at Ch. 4 Team Tests Problems 5-44 to 5-47 Problems 5-54 to 5-61 Conclusion Homework: Problems 5-48 to 5-53 AND 5-62 to 5-67 Ch. 4 Individual Test Friday –Study! [If you know you’re not going to be here due to extenuating circumstances, you must see me ahead of time to take the test.] 193

194 5-47: Learning Log Title: The Meaning of Solution, Part 2 What does the solution to an equation or a system of equations mean? What does a solution to a one variable equation look like on a graph? Algebraically? What does the solution to a system of equations look like on a graph? Algebraically? 194 194 194

195 5-54 -3 2

196 Day 42: March 29th Objective: Extend what was learned about solving systems of equations graphically to solving systems of inequalities. THEN Apply linear inequalities to solve a problem. HW Check and Correct (in red) Quickly! Problems 5-57 to 5-61 Problems 5-75 to 5-77 Conclusion Homework: Problems 5-68 to 5-74 Ch. 4 Individual Test Tomorrow –STUDY! 196

197 Solving a 1 Variable Inequality
Represent the solutions to the following inequality algebraically and on a number line. Closed or Open Dot(s)? Find the Boundary Test Every Region x Change inequality to equality Pick a point in each region Solve x = -4 x = 0 x = 3 Substitute into Original 9 ≤ 3 -3 ≤ 3 30 ≤ 24 False True False Shade True Region(s) Write Inequality Plot Boundary Point(s)

198 Solving a System of Inequalities
Graphically represent the solutions to the following system of inequalities: Solid or Dashed? Find the Boundaries Plot points for the equalities one at a time Test Every Region Find which side to shade for each inequality (0 ,0) (0 ,0) 0 ≥ -3 0 < 3 True True Shade the Feasible Region

199 5-60 199 199

200 5-61: Learning Log Title: The Meaning of Solution, Part 3 What does the solution to an equation or a system of equations mean? What does a solution to a one variable equation look like on a graph? Algebraically? What does the solution to a system of equations look like on a graph? Algebraically? What does the solution to an inequality or a system of inequalities mean? What does a solution to a one variable inequality look like on a graph? Algebraically? What does the solution to a system of inequalities look like on a graph? Algebraically? 200 200 200

201 Day 43: March 30th Objective: Assess Chapter 4 in an individual setting. Silence your cell phone and put it in your school bag (not your pocket) Get a ruler, pencil/eraser, and calculator out First: Calculator Portion…put calculator away when finished Second: Non-Calculator Portion (ask for it) Third: Check your work & hand the test in to Ms. Katz Fourth: Correct last night’s homework Homework: Problems 5-79 to 5-86 Enjoy your week away from school! 201

202 Day 44: April 10th Objective: Apply linear inequalities to solve a problem. HW Check and Correct (in red) Quickly! Problem 5-77 Start Problems 5-87 to [Graded Teamwork] Conclusion Homework: Problems 5-89 to 5-95 You will have another ½ hour tomorrow before you must hand in the good copy of your graded teamwork – you may want to work on it a little tonight Midterm (Ch. 5 Individual Test) Friday 202

203 x: Number of cars built y: Number of trucks built Define the Variables
Does it matter?

204 Cover-Up Method Plot : -2x + 5y = -10 Find the intercepts X Y -2 5

205 Constraints and Feasible Region
Non-Negative Constraints Cars: x-axis Trucks: y-axis Wheels: Seats: Gas Tanks: Profit Equation: Vertices of the Boundary P = c + 2t Graph the System: True Test Point (0,0) True Feasible Region True

206 Critical Points and Conclusion
Test Every Critical Point in the Profit Equation: P = c + 2t CONCLUSION: Otto should build 3 cars and 4 trucks for $11.

207 Day 45: April 11th Objective: Apply linear inequalities to solve a problem. THEN Learn how to simplify algebraic fractions. HW Check and Correct (in red) Quickly! Review Chapter 4 Test Work on Graded Teamwork (1/2 hour) Notes: Simplify Rational Expressions Rational Expressions 1 (Odds) Conclusion Homework: Problems 5-97 to 5-102 Do EVENS from Classwork Worksheet Finish team project – ready to hand in at 11:27 am Midterm (Ch. 5 Individual Test) Friday 207

208 Simplifying Rational Expressions
Simplify the following expressions:

209 Simplifying Rational Expressions
A fellow student simplifies the following expressions: Which simplification is correct? Substitute two values of x into each to justify your answer. MUST BE MULTIPLICATION!

210 Simplifying Rational Expressions
Can NOT reduce since everything does not have a common factor and it’s not in factored form Simplify: Factor Completely CAN reduce since the top and bottom have a common factor

211 Day 46: April 12th Objective: Understand how to multiply and divide rational expressions. THEN Understand how to add and subtract rational expressions and continue to learn how to simplify rational expressions. HW Check and Correct (in red) Quickly! & Hand-in Project Notes: Multiply/Divide Rational Expressions Rational Expressions 2 (Odds) Notes: Add/Subtract Rational Expressions Rational Expressions 3 (Odds) Chapter 5 Closure Homework: EVENS from Rational Expressions 2 & 3 (You can show me this HW tomorrow or Monday…but you will have another worksheet over the weekend) Midterm (Ch. 5 Individual Test) TOMORROW 211

212 Multiplying and Dividing Fractions
Multiply Numerators Multiply Denominators Divide: Multiply by the reciprocal (flip) Remember to Simplify!

213 Simplifying Rational Expressions
Half the work is done! Combine Rewrite Reduce

214 Simplifying Rational Expressions
Turn it into a multiplication problem Factor Factor Completely Reduce

215 Adding and Subtracting Fractions
Addition: Subtraction: Least Common Denominator (if you can find it) Common Denominator Subtract the Numerators Add the Numerators Remember to Simplify if Possible!

216 Add/Subtract Rational Expressions
Simplify: Same denominator! Half the work is done! CAREFUL with subtraction! Combine Like Terms Make sure it can’t be simplified more

217 Add/Subtract Rational Expressions
Simplify: Find a Common Denominator Combine Like Terms

218 Add/Subtract Rational Expressions
Simplify: Find a common denominator Distribute numerators but leave the denominators factored CAREFUL with subtraction Combine like Terms

219 Add/Subtract Rational Expressions
Simplify: Factor to find a Smaller Common Denominator Make sure it can’t be simplified beforehand

220 Add/Subtract Rational Expressions
Simplify: Factor to find a Smaller Common Denominator Make sure it can’t be simplified more

221 Day 47: April 13th Objective: Assess Chapters 1-5 in an individual setting. Silence your cell phone and put it in your school bag (not your pocket) Get a ruler, pencil/eraser, and calculator out First: Calculator Portion…put calculator away when finished Second: Non-Calculator Portion (ask for it) Third: Check your work & hand the test in to Ms. Katz Fourth: Correct last night’s homework Homework: Rational Expressions 4 (ALL) (And finish 2&3 Evens if you haven’t done so already) 221

222 Day 48: April 16th Objective: Learn to find rules that “undo” functions, and develop strategies to justify that each rule undoes the other. Graph functions along with their inverses and make observations about the relationships between the graphs. THEN Introduce the term “inverse” to describe undo rules. Also, graph the inverse of a function by reflecting it across the line of symmetry, and write equations for inverses. HW Check and Correct (in red) Quickly! Answer any questions about Rational Expressions Problems 6-1 to 6-6 Start Problems 6-16 to 6-25 Homework: Problems 6-7 to 6-15 AND 6-26 to 6-32 222

223 Guess my Number -70 Five Seven Eight -11 Three 3 and -3 Three and…
I’m thinking of a number that… When I… I get… My number is… Add four to my number AND Multiply by ten -70 Double my number Add four Divide by two Five Square my number Add three Add one Seven Subtract six Take the square root Eight -11 Three 3 and -3 Three and… Eleven

224 Only works when there is one x!
“Undo” Rule 1st Step 2nd Step 3rd Step p(x) p -1 (x) Start Add 3 Cube Multiply 2 Divide 2 Cube Root Subtract 3 Only works when there is one x!

225 Day 49: April 17th Objective: Introduce the term “inverse” to describe undo rules. Also, graph the inverse of a function by reflecting it across the line of symmetry, and write equations for inverses. HW Check and Correct (in red) Quickly! Wrap-Up Problems 6-5 to 6-6 Problems 6-16 to 6-25 Conclusion Homework: Problems 6-33 to 6-37 225

226 Tables and Graphs of Inverses
Switch x and y Original Inverse X Y 25 2 16 6 4 10 14 18 20 X Y X Y 25 16 2 4 6 10 14 18 20 Switch x and y (0,25) (20,25) (16,18) (2,16) (18,16) (4,14) (0,10) (6,4) (16,2) (14,4) (4,6) (10,0) Function Non-Function Line of Symmetry: y = x

227 6-6: Learning Log Title: Finding and Checking Undo Rules What strategies did your team use to find undo rules? How can you be sure that the undo rules you found are correct? What is another name for “undo?” How do the tables of a rule and an undo-rule compare? Graph? 227 227 227 227

228 Day 50: April 18th Objective: Introduce the term “inverse” to describe undo rules. Also, graph the inverse of a function by reflecting it across the line of symmetry, and write equations for inverses. THEN Use ideas of switching x- and y-values to learn how to find an inverse algebraically. Also, learn about compositions of functions and use compositions f(g(x)) and g(f(x)) to test algebraically whether two functions are inverses of each other. HW Check and Correct (in red) Quickly! Problems 6-22 to 6-25 and Slides Problems 6-38 to 6-42 Conclusion Homework: Problems 6-44 to 6-53 228

229 Inverse Notation Original function Inverse function

230 ± The Rule for an Inverse 1st Step 2nd Step 3rd Step 4th Step p(x)
Start Add 2 Square Multiply 3 Subtract 6 Square Root Add 6 Divide 3 Subtract 2

231 Vertical Line Test If a vertical line intersects a curve more than once, it is not a function. Use the vertical line test to decide which graphs are functions.

232 Horizontal Line Test If a horizontal line intersects a curve more than once, the inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function.

233 Restricted Domain Inverse Function Inverse
Find the inverse relation of f below: Inverse Function Inverse

234 Algebraically Finding an Inverse
Find the inverse of the following: Switch x and y Solve for y Do not write y-1 Make sure to check with a table and graph on the calculator.

235 Algebraically Finding an Inverse
Find the inverse of the following: Switch x and y Solve for y Do not write y-1 Because x2=9 has two solutions: 3 & -3 Make sure to check with a table and graph on the calculator.

236 Algebraically Finding an Inverse
Find the inverse of the following: Switch x and y Really y = Solve for y Make sure to check with a table and graph on the calculator.

237 Algebraically Finding an Inverse
Only Half Parabola Find the inverse of the following: Switch x and y Really y = Solve for y Restrict the Domain! Full Parabola (too much) x=3 Make sure to check with a table and graph on the calculator.

238 Day 51: April 19th Objective: Use ideas of switching x- and y-values to learn how to find an inverse algebraically. Also, learn about compositions of functions and use compositions f(g(x)) and g(f(x)) to test algebraically whether two functions are inverses of each other. THEN Apply strategies for finding inverses to parent graph equations. HW Check and Correct (in red) Quickly! Finish Problems 6-38 to 6-42 Problems 6-54 to 6-58 Conclusion Homework: Problems 6-59 to 6-66 Ch. 6 Team Test Thursday 238

239 Composition of Functions
Substituting a function or its value into another function. Second g f First (inside parentheses always first) OR

240 Composition of Functions
Let and Find: Equivalent Statements Our text uses the first one Plug x=1 into g(x) first Plug the result into f(x) last

241 Composition of Functions
Let and Find: Plug the result into g(x) last Plug x into f(x) first

242 Inverse and Compositions
In order for two functions to be inverses: AND

243 Day 52: April 20th Objective: Apply strategies for finding inverses to parent graph equations. THEN Define the term logarithm as the inverse exponential function or, when y=bx, “y is the exponent to use with base b to get x.” HW Check and Correct (in red) Quickly! Finish Problems 6-56 to 6-58 Problems 6-67 to 6-71 Conclusion Homework: Problems 6-72 to 6-80 Ch. 6 Team Test Thursday 243

244 Silent Board Game

245 Silent Board Game

246 Logarithm and Exponential Forms
Logarithm Form 5 = log2(32) Base Stays the Base Input Becomes Output Logs Give you Exponents 25 = 32 Exponential Form

247 Examples Write each equation in exponential form log125(25) = 2/3
log8(x) = 1/3 Write each equation in logarithmic form If 64 = 43 If 1/27 = 3x 1252/3 = 25 81/3 = x log4(64) = 3 log3(1/27) = x

248 Day 53: April 23rd Objective: Develop methods to graph logarithmic functions with different bases. Rewrite logarithmic equations as exponential equations, and find inverses of logarithmic functions. THEN Look into the base of the log key on the calculator. Extend knowledge of general equations for parent functions to transform the graph of y = log(x). HW Check and Correct (in red) Quickly! Check Problems 6-70 to 6-71 Logarithms and Graphs Packet (Extra Visual) Problems 6-93 and 6-95 Conclusion Homework: Problems 6-84 to 6-92 AND 6-96 to 6-105 Ch. 6 Team Test Thursday Ch. 6 Individual Test Tuesday 248

249 Inverse of an Exponential Equation
Original Inverse OR Logs give you exponents!

250 Definition of Logarithm
The logarithm base a of b is the exponent you put on a to get b: i.e. Logs give you exponents! a > 0 and b > 0

251 6-71: Closure 2 4 7 1.2 w + 3

252 Day 54: April 25th Objective: Look into the base of the log key on the calculator. Extend knowledge of general equations for parent functions to transform the graph of y = log(x). HW Check and Correct (in red) Quickly! Wrap-up/Recap Logs and Graphs exploration Problems 6-93 and 6-95 Review Midterm Introduction to Chapter 7 Conclusion Homework: Problems to (Skip 116, 118) Change 113 to the square root of 7-x Ch. 6 Team Test Tomorrow Ch. 6 Individual Test Tuesday 252

253 6-83: Learning Log What is the general shape of the graph?
Title: The Family of Logarithmic Functions What is the general shape of the graph? What happens to the value of y as x increases? How is the graph related to the exponential graph? What is the Domain? Range? Why is the x-intercept always (1,0)? Why is the line x=0 (y-axis) always an asymptote? Why is there no horizontal asymptote? How does the graph change if b changes? What does the graph look like when 0<b<1? What does the graph look like when b=1? What does the graph look like when b>1? 253 253 253

254 Common Logarithm Ten is the common base for logarithms, so log(x) is called a common logarithm and is shorthand for writing log10(x). You read this as “the logarithm base 10 of x.” Our calculator has the button log . It doesn’t have the subscript 10 because it stands for the common logarithm: log10100 = log100

255 Logarithmic Function Parent Equation Graphing Form

256 Example: Logarithmic Transformation: Shift the parent graph three units to the right and two units up. Transformation: New Equation: y = 2 x = 3

257 Day 55: April 26th Objective: Assess Chapter 6 in a team setting. THEN Create and use a model to locate points in 3-D space, and plot points in 3-D on isometric paper. HW Check and Correct (in red) Quickly! Problem from HW should not have had a “square” on the 7 minus x… Chapter 6 Team Test Introduction to Chapter 7 Start Problems 7-1 to 7-7 Conclusion Homework: Problems CL6-121 to AND 7-8 to 7-15 [Check this assignment w/Ms. Katz before leaving today] Ch. 6 Individual Test Tuesday 257

258 Day 56: April 27th Objective: Create and use a model to locate points in 3-D space, and plot points in 3-D on isometric paper. THEN Graph planes. *NEW SEATS* HW Check and Correct (in red) Quickly! Problems 7-1 to 7-7 Problems 7-16 to 7-20 Conclusion Homework: Problems 7-8 to 7-15 AND 7-21 to 7-28 Ch. 6 Individual Test Tuesday Start evaluating your textbook…if your cover is torn/missing or there is other significant damage, you owe $19 to replace it. Please do not make a mess of it with tape – if you think it can be repaired, see Ms. Katz. Otherwise, bring cash or check by the time we finish Chapter 7. 258

259 Plotting Points in xyz-Space
( 2 , 3 , 5 ) x y Link

260 Plotting Planes in xyz-Space

261 Day 57: April 30th Objective: Graph planes. THEN Investigate the graphs of systems of equations with three variables. Find the points that lie on two planes simultaneously. HW Check and Correct (in red) Quickly! Review Ch. 5 Project and Ch. 6 Team Test Finish Problems 7-18 to 7-20 Problems 7-29 to 7-33 Start Problems 7-43 to 7-48 and 7-49 Conclusion Homework: Problems 7-39 to 7-42 & STUDY! Ch. 6 Individual Test Tomorrow Start evaluating your textbook…if your cover is torn/missing or there is other significant damage, you owe $19 to replace it. Please do not make a mess of it with tape – if you think it can be repaired, see Ms. Katz. Otherwise, bring cash or check by the time we finish Chapter 7. 261

262 7-20: x=4 in Different Dimensions
One Dimension Two Dimensions Three Dimensions Point Line Plane

263 Day 58: May 1st Objective: Assess Chapter 6 in an individual setting.
Silence your cell phone and put it in your school bag (not your pocket) Get a ruler, pencil/eraser, and calculator out First: Calculator Portion…put calculator away when finished Second: Non-Calculator Portion (ask for it) Third: Check your work & hand the test in to Ms. Katz Fourth: Correct last night’s homework Homework: Problems 7-34 to 7-38 AND 7-50 to 7-59 Start evaluating your textbook…if your cover is torn/missing or there is other significant damage, you owe $19 to replace it. Please do not make a mess of it with tape – if you think it can be repaired, see Ms. Katz. Otherwise, bring cash or check by the time we finish Chapter 7. 263

264 Day 59: May 2nd Objective: Develop an algebraic strategy to solve systems of three equations with three variables. Also, determine the different ways three planes can intersect, and investigate the graphs of 3-D systems. THEN Find the equation of a quadratic function y=ax2+bx+c that passes through three given points when graphed. HW Check and Correct (in red) Quickly! Finish Problems 7-43 to 7-48 and 7-49 Problems 7-60 to 7-68 Conclusion Homework: Problems 7-71 to 7-86 Start evaluating your textbook…if your cover is torn/missing or there is other significant damage, you owe $19 to replace it. Please do not make a mess of it with tape – if you think it can be repaired, see Ms. Katz. Otherwise, bring cash or check by the time we finish Chapter 7. 264

265 Solving a 3 Variable System
Solve the system: 2. Solve the system + + + 1. Use Elimination to write a 2-Variable System 3. Solve for the 3rd Variable Must be the same 2 variables! 4. Solution:

266 Solving a 3 Variable System
Solve the system: 2. Solve the new system 1 You must multiply to eliminate 2 + 3 1. Use Elimination to write a 2-Variable System Multipliy 2nd by 2 Multiply 2nd by 3 3. Solve for the 3rd Variable 1 2 + 2 + 3 4. Solution:

267 Two Forms of a Quadratic
y = ax2 + bx + c y = a(x – h)2 + k Standard Form Parabola a is the stretch factor a tells whether it opens up/down Can be put into factored form Use the quadratic formula c is the y-intercept Graphing/General Form (h,k) is the vertex

268 Writing a Contextual 3 Variable System
Suppose the graph of a quadratic function passes through the points (1,0), (2,5), and (3,12). Algebraically find the quadratic equation. 1. Use the Standard Quadratic Form: x y 2. Substitute each Point into the Equation: 3. Simplify the Equations:

269 Solving a Contextual 3 Variable System
Solve the system: 2. Solve the new system 1 Eliminate the “c” first! 2 + 3 1. Use Elimination to write a 2-Variable System 1 1 3. Solve for the 3rd Variable 2 + + 3 4. Subsititue into the Standard Form:

270 Day 60: May 3rd Objective: Develop the Power Property of Logs and use it to develop an efficient method to solve exponential equations in ax=b form. THEN Learn the Product and Quotient Properties of logs and how to rewrite equations with different bases. HW Check and Correct (in red) Quickly! Wrap-Up Problems 7-64 to 7-68 Problems 7-87 to 7-93 Start Problems to 7-109 Conclusion Homework: Problems 7-94 to 7-102 270

271 Day 61: May 4th Objective: Learn the Product and Quotient Properties of logs and how to rewrite equations with different bases. THEN Develop strategies for finding the equation of an exponential function given two points and an asymptote. HW Check and Correct (in red) Quickly! Problems to 7-110 Problems to 7-126 Conclusion Homework: Problems to AND to 7-136 (These are long…leave yourself enough time!) 271

272 Power Property of Logs

273 Solving Equations with the Power Property of Logs

274 The Change of Base Formula
For a and b greater than 0 AND b≠1.

275 Properties of Logarithms
Power Property: Product Property: Quotient Property:

276 Day 62: May 7th Objective: Develop strategies for finding the equation of an exponential function given two points and an asymptote. THEN Apply knowledge of exponential functions to solve a murder mystery. THEN Add, subtract, and start to multiply matrices. HW Check and Correct (in red) Quickly! Review Chapter 6 Individual Test Problems to 7-126 Problem 7-137 Notes: Matrices Conclusion Homework: Problems to AND to (CW) 276

277 System of Exponential Equations
Find an exponential function that passes through (3,12.5) and (4,11.25) and has a horizontal asymptote of y = 10. Asymptote c=10 Substitute into twice: Larger exponent first Substitute into either equation to find a – 10 – 10 – 10 – 10 Rewrite into y=abx Divide #s ÷ Subtract Exponents Warning: This is not addressed a lot in the homework but will be assessed.

278 Day 63: May 8th Objective: Apply knowledge of exponential functions to solve a murder mystery. THEN Add, subtract, and start to multiply matrices. THEN Use matrix multiplication to solve problems. HW Check and Correct (in red) Quickly! Finish Problem 7-137 Notes: Matrices Problems to 7-154 Start Problems to 7-169 Conclusion Homework: Problems to AND to 7-174 (Do more if you have time) Chapter 7 Team Test Thursday Is your book cover torn? Is your book in poor condition? Bring cash/check ($19) so that I can replace it. Please DO NOT attempt to tape it with white tape! See me if you think it can be repaired. 278

279 Matrix A matrix M is an array of cell entries (mrow,column) and it must have rectangular dimensions (Rows x Columns). Example: A arow,column 4 A 3 15x 3x4 Dimensions:

280 Scalar Multiplication
Every entry in the matrix is multiplied by the number outside the matrix (scalar). Example:

281 Matrix Addition/Subtraction
IF the matrices have the same dimensions, add or subtract corresponding cell entries. Examples: b+h

282 Matrix Addition/Subtraction
Perform the indicated operation: The matrices MUST have the same dimensions!

283 Matrix Multiplication
1Multiply the elements of each row of the first matrix by the elements of each column in the second matrix. 2Add the products. 3The answer goes into arow of 1st, column of 2nd. 3x2 2x3 a1,1 a1,2 1 a2,1 a2,2 2 1 2 15 16 2x2 20 27

284 Matrix Multiplication
Can we multiply these… 5x1 2x3 2x2 3x4 ? No No 3x2 # of columns in 1st MUST be the same as # of rows in 2nd! 1x3 Yes

285 Day 64: May 9th Change the W’s to E’s!
Objective: Use matrix multiplication to solve problems. THEN Use a graphing calculator to perform operations with matrices. HW Check and Correct (in red) Quickly! Problem 7-154 Problems to 7-169 Start Problems to 7-184 Conclusion Homework: Problems to AND to 7-193 Chapter 7 Team Test Tomorrow Is your book cover torn? Is your book in poor condition? Bring cash/check ($19) so that I can replace it. Please DO NOT attempt to tape it with white tape! See me if you think it can be repaired. Change the W’s to E’s! 285

286 Matrix Multiplication with a Context
Cars Trucks Wheels Seats Gas Tanks Bull’s Eye Order Cars B A JC Nickels Department Store Order Trucks Wheels Seats Gas Tanks Bull’s Eye Total Order JC Nickels Department Store Total Order

287 Matrices from and 7-172

288 Day 65: May 10th Objective: Assess Chapter 7 in a team setting. THEN Use a graphing calculator to perform operations with matrices. HW Check and Correct (in red) Quickly! Chapter 7 Team Test Finish Problems to 7-184 Conclusion Homework: Finish Problems to AND to 7-208 Chapter 7 Individual Test next week (Thursday?) Is your book cover torn? Is your book in poor condition? Bring cash/check ($19) so that I can replace it. Please DO NOT attempt to tape it with white tape! See me if you think it can be repaired. 288

289 Day 66: May 11th Objective: Use a graphing calculator to perform operations with matrices. THEN Write systems of equations as matrix equations. Find the identity element for a matrix and consider inverses for matrices. HW Check and Correct (in red) Quickly! Wrap-Up Problems to 7-184 Problems to 7-199 Conclusion Homework: Finish Problems to AND to 7-226 Chapter 7 Individual Test next week (Thursday?) Is your book cover torn? Is your book in poor condition? Bring cash/check ($19) so that I can replace it. We will be trading books TUESDAY – bring yours! Please DO NOT attempt to tape it with white tape! See me if you think it can be repaired. 289

290 Order in Matrix Multiplication Matters

291 Matrix Multiplication
The dimensions of a product of matrices are the # of rows of the first matrix by the # of columns of the second matrix. 179 (b) 3 3x3 3x3 3 3x3 3 3 180 (a) 1 1x3 3x3 3 1 1x3 3 182 2x2 2 2x2 2 2x2 2 2 183 (b) 3x3 3 3x1 1 3x1 3 1

292 The Race

293 Day 67: May 14th Objective: Write systems of equations as matrix equations. Find the identity element for a matrix and consider inverses for matrices. THEN Solve systems of equations using matrices and graphing calculators. THEN Introduce a simplified method of finding the vertex of a quadratic function. HW Check and Correct (in red) Quickly! Finish Problems to 7-199 Problems to 7-216 Vertex Simplified Notes Conclusion – Recap Chapter 7 Topics Homework: Problems to AND CL7-228 to 7-238 BRING TEXTBOOK (OR $19) FROM HOME! Chapter 7 Individual Test (Thursday?) Is your book in poor condition? Bring cash/check ($19) so that I can replace it. We will be trading books TUESDAY – bring yours! Please DO NOT attempt to tape it with white tape! 293

294 I Must be a square matrix
Identity Matrix The product of a square matrix A and its identity matrix I, on the left or the right, is A. AI = IA =A General Form: I Must be a square matrix

295 Identity Matrix Example
The identity matrix must be the same dimensions with 0’s in every cell except for 1’s in the main diagonal Must be a square matrix The same!

296 (A Must be a square matrix)
Inverse Matrix The product of a square matrix A and its inverse matrix A-1, on the left or the right, is the identity matrix I. AA-1= A-1A =I (A Must be a square matrix) How do we find the Inverse Matrix:

297 Converting a System of Equations to a Matrix Equation
Make sure the equations are in alphabetical order Identify all of the coefficients to the variables Coefficient Matrix Variable Matrix Constant Matrix

298 Solving a System of Equations with Matrices
Solve: Identify all of the coefficients to the variables Make sure the equations are in alphabetical order and that every variable is in each equation Coefficient Matrix “A” Variable Matrix “X” Constant Matrix “B”

299 Solving a Systems of Equations with Matrices
Continued… A X B Which Order is Correct? Multiply by the inverse of A to isolate the variable matrix 3x3.3x1 A-1 A X A-1 B 3x1.3x3 OR A-1 A X B A-1

300 Solving a System of Equations with Matrices
Continued… A X B Step 1: Store Matrix A and B in your calculator Multiply by the inverse of A to isolate the variable matrix A-1 A X A-1 B Step 2: Enter this in your calculator to solve the system THUS: You do not need to calculate the inverse matrix! If… Then…

301 y = -3x2 + 6x – 7 Finding a Vertex Find the coordinates of the vertex.
Use the following equation to answer the questions below: y = -3x2 + 6x – 7 Find the coordinates of the vertex. Write the equation in graphing form.

302 Vertex Simplified x y is plug and chug with x
If f(x) = ax2 + bx + c, then the vertex is: x Opposite of b y is plug and chug with x

303 Example y = -3x2 + 6x – 7 Find the coordinates of the vertex.
Use the following equation to answer the questions below: y = -3x2 + 6x – 7 Find the coordinates of the vertex. Write the equation in graphing form.


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