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Warm Up

Warm Up. Unit 3: In Review. Unit 3 Test A. 1. Function Notation to Represent & Solve Equations Use function notation Combining Functions (using mathematical operations) Recognize solutions on a graph Know domain and range 2. Interpret and Analyze Functions Using Different Representations

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Warm Up

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  1. Warm Up

  2. Unit 3: In Review

  3. Unit 3 Test A 1. Function Notation to Represent & Solve Equations • Use function notation • Combining Functions (using mathematical operations) • Recognize solutions on a graph • Know domain and range 2. Interpret and Analyze Functions Using Different Representations 3. Build New Functions from Existing Functions

  4. Unit 3: Lesson 1 Unit Outline Title: Linear and Exponential Functions Name of Lesson: Function Notation to Represent & Solve Equations Standards: MCC9-12.A.REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). MCC9-12.F.IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). MCC9-12.F.IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. MCC9-12.A.REI.11Explain why the x-coordinates of a the points where the graphs of the equations y= f(x) and y = g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions,, make where f(x) and /or g(x) are linear and exponential functions. MCC9-12.F.BF.1bCombine standard functiontypes using arithmetic operations. (Lesson 4)

  5. MCC9-12.F.IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MCC9-12.F.IF.1Understand that a function from one set (called the domain) to another set (called the range) Recall: We are use to writing: y = mx + b Now we know: f(x) = mx + b Mapping 1 2 3 7 8 9 Ordered Pairs (1, 9) (2, 8) (3, 7) This replaces the y 1 2 3 9 8 7 Table Domain: Range: {1, 2, 3} {7, 8, 9}

  6. Unit 3: Lesson 1 Unit Outline Title: Linear and Exponential Functions Name of Lesson: Function Notation to Represent & Solve Equations Standards: MCC9-12.A.REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). MCC9-12.F.IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). MCC9-12.F.IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. MCC9-12.A.REI.11Explain why the x-coordinates of a the points where the graphs of the equations y= f(x) and y = g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions,, make where f(x) and /or g(x) are linear and exponential functions. MCC9-12.F.BF.1bCombine standard functiontypes using arithmetic operations. (Lesson 4)

  7. MCC9-12.F.BF.1bCombine standard functiontypes using arithmetic operations.(Lesson 4) Given: f(x) = 2x + 3 g(x) = 3x + 2 h(x) = x2 – 2x + 1 Find the following. 1. f(0) 2. h(-3) 3. f(x) + g(x) 4. h(x) – f(x) 5. h(x)f(x) = x2 – 2x + 1 (2x + 3) = x2– 2x + 1 – 2x – 3 = x2 – 4x – 2 h(-3) = (-3)2 – 2(-3) + 1 = 9 + 6 +1 h(-3) = 16 f(0) = 2(0) + 3 = 0 + 3 f(0) = 3 f(x)+ g(x) = 2x + 3 + 3x + 2 = 2x + 3x + 5 h(x)– f(x) f(x)+ g(x)

  8. Unit 3: Lesson 1 Unit Outline Title: Linear and Exponential Functions Name of Lesson: Function Notation to Represent & Solve Equations Standards: MCC9-12.A.REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). MCC9-12.F.IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). MCC9-12.F.IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. MCC9-12.A.REI.11Explain why the x-coordinates of a the points where the graphs of the equations y= f(x) and y = g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions,, make where f(x) and /or g(x) are linear and exponential functions. MCC9-12.F.BF.1bCombine standard functiontypes using arithmetic operations. (Lesson 4)

  9. MCC9-12.A.REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). MCC9-12.A.REI.11Explain why the x-coordinates of a the points where the graphs of the equations y= f(x) and y = g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions,, make where f(x) and /or g(x) are linear and exponential functions. One Solution: (-5, -4) Infinitely many One Solution: (8, 8) No Solution

  10. Unit 3 Test A 1. Function Notation to Represent & Solve Equations • Use function notation • Combining Functions (using mathematical operations) • Recognize solutions on a graph • Know domain and range 2. Interpret and Analyze Functions Using Different Representations • Know ALL your vocab and be able to write out your understanding • Graph Linear and Exponential Graphs 3. Build New Functions from Existing Functions • Transformations • Even and Odd Symmetry

  11. Unit 3: Lesson 2 Unit Outline Title: Linear and Exponential Functions Topic Title: Interpret and Analyze Functions Using Different Representations Standards: MCC9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior. MCC9-12.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. MCC9-12.F.IF.7aGraph linear functions and show intercepts, maxima, and minima. MCC9-12.FIF.7eGraph exponential functions, showing intercepts and end behavior. MCC9-12.F.IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal description). Vocabulary Graphing

  12. MCC9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior. Can you tell me…. Where are these graphs Increasing? (-∞, ∞) (-∞, ∞) Where are these graphs Decreasing? As x  -∞, f(x)  0 As x  ∞, f(x)  ∞ As x  -∞, f(x)  -∞ As x  ∞, f(x)  ∞ Are they positive, negative? Symmetric? End Behavior?

  13. MCC9-12.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. MCC9-12.F.IF.7aGraph linear functions and show intercepts, maxima, and minima. MCC9-12.FIF.7eGraph exponential functions, showing intercepts and end behavior. MCC9-12.F.IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal description). Practice Graphing at home! Next point…. transformations

  14. Unit 3 Test A 1. Function Notation to Represent & Solve Equations • Use function notation • Combining Functions (using mathematical operations) • Recognize solutions on a graph • Know domain and range 2. Interpret and Analyze Functions Using Different Representations • Know ALL your vocab and be able to write out your understanding • Graph Linear and Exponential Graphs 3. Build New Functions from Existing Functions • Transformations • Even and Odd Symmetry

  15. Unit 3-Lesson 4 Unit Outline Title: Linear and Exponential Functions Topic Title: Build New Functions from Existing Functions Standards: MCC9-12.F.BF.3Identify the effect on the graph of replacing f(x) by f(x)+k, k f(x), and f(x+k) for specific values of k(both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation for the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. [transformations]

  16. MCC9-12.F.BF.3Identify the effect on the graph of replacing f(x) by f(x)+k, k f(x), and f(x+k) for specific values of k(both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation for the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Even Functions Odd Functions Symmetric across the origin f(x) = -f(x) All ODD exponents f(x) = x7 + 2x3 – x Definition: Equations: Algebraic Rule: Algebra Example: Graph Example: Symmetric across the y-axis f(x) = f(-x) All EVEN exponents f(x) = 3x8 + 2x2 + 1

  17. MCC9-12.F.BF.3Identify the effect on the graph of replacing f(x) by f(x)+k, k f(x), and f(x+k) for specific values of k(both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation for the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. y = -a(x – h)n + k y = -a(x – h)n + k y = -a(x – h)n + k y = -a(x – h)n + k y = -a(x – h)n + k We started with: k: Vertical Shift + k, moves up - k, moves down Then we learned how to relate these rules to exponential graphs h: Horizontal Shift + h, moves left - h, moves right a: Vertical Stretch/Shrink a > 1, stretches the graph upward 0<|a|<1, shrinks the graph outward – : reflect over the x-axis

  18. MCC9-12.F.BF.3Identify the effect on the graph of replacing f(x) by f(x)+k, k f(x), and f(x+k) for specific values of k(both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation for the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Now let’s look at Exponentials: k: Vertical Shift + k, moves up - k, moves down bx – a· + k + h – h: Horizontal Shift + h, moves left - h, moves right Study your formula! -a·b-x+h+ k a: Vertical Stretch/Shrink a > 1, stretches the graph upward 0<|a|<1, shrinks the graph outward y = -a(x – h)n + k – : reflect over the x-axis – : reflect over the y-axis

  19. Unit 3 Test A 1. Function Notation to Represent & Solve Equations • Use function notation • Combining Functions (using mathematical operations) • Recognize solutions on a graph • Know domain and range 2. Interpret and Analyze Functions Using Different Representations • Know ALL your vocab and be able to write out your understanding • Graph Linear and Exponential Graphs 3. Build New Functions from Existing Functions • Transformations • Even and Odd Symmetry

  20. Unit 3 Test B • Build a function that models a relationship between two quantities • Determine the sequence • Write the equation for the sequence • Solve the sequence • Construct and Compare Linear and Exponential Models and Solve Problems • Compare Linear and Exponential Graphs • Rate of Change *using graphs and tables We have already started this! This too, more word problems

  21. Unit 3-Lesson 3 Unit Outline Title: Linear and Exponential Functions Topic Title: Build a function that models a relationship between two quantities Standards: MCC9-12.F.BF.1 Write a function that describes a relationship. MCC9-12.BF.1aDetermine an explicit expression, a recursive process, or steps for calculation from a context. MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. MCC9-12.F.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations and translate between the two forms.

  22. Unit 3-Lesson 5 Unit Outline Title: Linear and Exponential Function Topic Title: Construct and Compare Linear and Exponential Models and Solve Problems Standards: MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. MCC9-12.F.LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential function grow by equal factors over equal intervals. MCC9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. (Exponential Functions) MCC9-12.F.LE.2Construct linear and exponential functions, (graphing) including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). MCC9-12.F.LE.3Observe using graphs and tables that quantity increasing exponentially eventually exceeds a quantity increasing linearly. MCC9-12.F.LE.5Interpret the parameters in a linear or exponential function in terms of a context.

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