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Honors Algebra 2 Spring 2012 Ms. Katz

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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

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Five Point Star

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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.

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**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

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Function Machines (a) 6 -16 7 121 11

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Function Machines (b) 64 8 -36 17 131065

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**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

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**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:

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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

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**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)

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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

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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.

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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.

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**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)

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**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.

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**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

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**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

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**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

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**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

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**Example: Inequalities**

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

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**Example: Inequalities**

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

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**Example: Inequalities**

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

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**Example: Inequalities**

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

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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

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**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

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**Multiple Representations**

Non-Algebraic Table Rule or Equation Graph Context Algebraic

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**Solving a System Algebraically**

Use the equations to solve the following system:

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**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

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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

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

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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

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**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?

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**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

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**(Multiple Values/Vary)**

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

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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

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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

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**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

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**Base raised to an exponent**

Exponential Notation BaseExponent Base raised to an exponent

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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:

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**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

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**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

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Exploration Simplify: =

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

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Example Simplify: =

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Exploration Simplify:

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**Power of a Power To find a power of a power, multiply the exponents.**

Stress same base 47

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Example Simplify: = 8s13t11

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Exploration Simplify:

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**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

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Example Simplify: = -288x7y20

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**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

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**A Negative Exponent A simplified expression has no negative exponents.**

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

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Example Simplify:

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Exploration Simplify: =

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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

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Example Simplify:

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Exploration Simplify:

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

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Example Simplify:

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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

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**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 …

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**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 …

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**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 …

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**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 …

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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

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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

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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

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Exponential Graph Notes: Horizontal Asymptote

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**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

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**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

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**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

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**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

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Rebound Ratio

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

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Discrete Graph The graph is made up of separate points.

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**Starting Height vs. Rebound Height**

y = mx + b y = mx Rebound Height Rebound Ratio Starting Height Starting Height

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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

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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

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**Bounce vs. Rebound Height**

Bounce Number Rebound Height y = abx Rebound Height Starting Height Rebound Ratio Bounce Number Discrete!

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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.

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**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

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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

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**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

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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

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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

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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

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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

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**Generator for a Percent Increase**

What is a 15% increase of 100? First Step: Second Step: . . 15.% = . 0.15 2 1 Multiplier

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**Generator for a Percent Decrease**

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

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

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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

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Exponential Function Time Initial Rate

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**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

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**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

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**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

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**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

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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

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**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

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Two Butt Cheeks When there is addition or subtraction:

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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

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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

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**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

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**Begin to Investigate Exponentials**

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

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**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?

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**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

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**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

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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

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**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

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**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:

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**Exponential Function Web**

Table Rule or Equation Graph Context

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**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

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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

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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

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**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

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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

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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

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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

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**Simple: Compound: Interest P = Principal Amount (original)**

r = rate ( % ÷ 100 ) t = time in years n = number of intervals

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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

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**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

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**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

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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

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**Exponential Function Web**

Table Rule or Equation Graph Context

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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).

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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

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Radical Property ONLY when a≥0 and b≥0

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**Exponents into Radical Notation**

bp/q = Generally b≥0

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**Evaluate the following without a calculator:**

Example Evaluate the following without a calculator:

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**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

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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

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**By the End of the Chapter…**

You will be able to easily sketch graphs similar to the following by just looking at the equations:

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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

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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

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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

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**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.

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Vertical Dilations Transform! Vertical Stretch Vertical Compression

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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

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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

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**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

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**–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

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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

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**Graphing Form to Standard Form**

Same a!

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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

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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

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**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

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**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

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**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.

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**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:

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**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!

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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

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**Function Transformations**

Family Parent Graphing Calculator Cubic Hyperbola Square Root Exponential

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**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

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

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**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

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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

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**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)

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**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

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**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!

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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

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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

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**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(

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**Absolute Value Function**

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

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**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

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Sleeping Parabola Parent: Graphing Form: Calculator:

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Circle Parent: Graphing Form: Calculator:

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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

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**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?

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**(0,0) (h,k) Equation for a Circle Example Graphing Form**

Center: Radius: Graphing Form (h,k) Center: Radius:

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**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!

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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

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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

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

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**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?

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**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.

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**Completing the Square Find the c that completes the square:**

x2 + 50x + c x2 – 22x + c x2 + 15x + c

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**Factoring a Completed Square**

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

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**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

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

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**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:

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**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:

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**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:

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**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

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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

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**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

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Extraneous Solutions

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**Calculator & Solving Equations**

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

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**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

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**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

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5-34

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**Solve the Following Algebraically**

OR

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**Solve the Following Algebraically**

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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

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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

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5-54 -3 2

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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

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**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)

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**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

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5-60 199 199

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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

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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

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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

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**x: Number of cars built y: Number of trucks built Define the Variables**

Does it matter?

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

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**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

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**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.

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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

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**Simplifying Rational Expressions**

Simplify the following expressions:

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**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!

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**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

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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

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**Multiplying and Dividing Fractions**

Multiply Numerators Multiply Denominators Divide: Multiply by the reciprocal (flip) Remember to Simplify!

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**Simplifying Rational Expressions**

Half the work is done! Combine Rewrite Reduce

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**Simplifying Rational Expressions**

Turn it into a multiplication problem Factor Factor Completely Reduce

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**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!

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**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

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**Add/Subtract Rational Expressions**

Simplify: Find a Common Denominator Combine Like Terms

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**Add/Subtract Rational Expressions**

Simplify: Find a common denominator Distribute numerators but leave the denominators factored CAREFUL with subtraction Combine like Terms

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**Add/Subtract Rational Expressions**

Simplify: Factor to find a Smaller Common Denominator Make sure it can’t be simplified beforehand

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**Add/Subtract Rational Expressions**

Simplify: Factor to find a Smaller Common Denominator Make sure it can’t be simplified more

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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

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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

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**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

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**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!

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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

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**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

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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

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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

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Inverse Notation Original function Inverse function

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**± 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

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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.

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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

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**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.

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**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.

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**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.

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**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.

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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

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**Composition of Functions**

Let and Find: Plug the result into g(x) last Plug x into f(x) first

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**Inverse and Compositions**

In order for two functions to be inverses: AND

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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

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Silent Board Game

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Silent Board Game

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**Logarithm and Exponential Forms**

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

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**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

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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

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**Inverse of an Exponential Equation**

Original Inverse OR Logs give you exponents!

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**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

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

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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

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**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

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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

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Logarithmic Function Parent Equation Graphing Form

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

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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

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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

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**Plotting Points in xyz-Space**

( 2 , 3 , 5 ) x y Link

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**Plotting Planes in xyz-Space**

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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

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**7-20: x=4 in Different Dimensions**

One Dimension Two Dimensions Three Dimensions Point Line Plane

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**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

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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

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**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:

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**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:

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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

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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

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Power Property of Logs

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**Solving Equations with the Power Property of Logs**

274
**The Change of Base Formula**

For a and b greater than 0 AND b≠1.

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**Properties of Logarithms**

Power Property: Product Property: Quotient Property:

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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.

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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

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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:

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**Scalar Multiplication**

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

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**Matrix Addition/Subtraction**

IF the matrices have the same dimensions, add or subtract corresponding cell entries. Examples: b+h

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**Matrix Addition/Subtraction**

Perform the indicated operation: The matrices MUST have the same dimensions!

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**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

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**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

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Matrices from and 7-172

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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

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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

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**Order in Matrix Multiplication Matters**

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**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

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The Race

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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

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**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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