Relationships between Inputs & Outputs

Relationships between Inputs & Outputs 2A I can use mapping to explain the concept of composite functions and how the connection between sets operates as the rule. I can break functions down into composite pieces and find the domains for composite functions including polynomial, radical, and rational functions 2B I can analyze functions to determine if they are classified as odd functions, even functions, or neither and explain the relationship between input and outputs 2C Given a graph, I can analyze relationships between inputs and outputs (composition) and find outputs by analyzing relationships (composite, translations, absolute value)

Non-Routine GP #2 LT 2A & 2C The suggested retail price of a new car is p dollars. The dealership advertized a factory rebate of $1200 and an 8% discount. Write a function R in terms of p giving the cost of the car after receiving the factory rebate. Write a function S in terms of p giving the cost of the car after the 8% discount. Does or yield the lower cost of the car? Explain.

Non-Routine #2 LT 2A & 2C Soln The suggested retail price of a new car is p dollars. The dealership advertized a factory rebate of $1200 and an 8% discount. • R = P - 1200 • S = 0.92P • = (0.92)(p-1200) = 0.92p-1200 0.92p-1200 <0.92 (p-1200) < Did you Match or Did you take another Pathway?

Describe the symmetry Describe symmetry

Describe the Relationship Approach Independently Create 2 Plans that might work by communicating with partner Approach Create Plans

II. Even and Odd Functions Definitions 1. Even functions: Let f(x) be a real-valued function. Then f(x) is even if the following relationship is true: f(-x) = f(x) for all x in the domain of f. a. Geometrically, the graph of an even function is symmetric with respect to the y-axis On your own, use the definition to sketch an example of a function that MIGHT be even. Convince your partner that your sketch satisfies the definition of an even function.

Visual Explain how the symmetry works

Even and Odd Functions 2. Odd functions: Let f(x) be a real-valued function. Then f(x) is odd if the following relationship is true: f(-x) = -f(x) a. Geometrically, the graph of an odd function has rotational symmetry with respect to the origin (y=x) Describe the resulting image when you place a pin at the origin and rotate the image 180 degrees

Visual Explain how the symmetry works

Use a function relationship to describe this situation

Why Eggs? An easter egg is a joke/visual gag/in-joke that a creator (typically the artist) has hidden in the artwork for viewers to find (just like an easter egg). They range from the not-so-obscure to the really obscure.

WHERE IS THE EASTER EGG?

Who Cares about Even/Odd Functions? • Purpose: To figure out symmetry relations between the pre-image/domain and image/range of a function • to help sketch quickly • to use as a short-cut when differentiating and integrating Calculus

Goal Problems #1 (LT 2B) Recall & Reproduction Routine Given f(x)= 1/x. Use the definitions to determine if f(x) is even, odd or neither? Odd function, Even function or Neither? Explain your reasoning. Write out approach, questions and areas of confusion

Active PracticeLearning Target 2B Note to Self in Margin of notes • My Personal Learning Goal during active practice. • My choices to ensure action levels up my learning today. • How will I know if my choices are working? When will I try the Non-Routine? When will I build my concept map?

Non-Routine Problem LT 2B • http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/performance-tasks/crickets.pdf • Need to print student sheets (pages 5-6). Problem will not fit here.

Goal Problems #2 LT 2B Recall & Reproduction Routine Find the coordinates of a second point on the graph given that: Function is even, (5, -1) Function is odd, (4, 9) Determine algebraically if the function is even, odd or neither: f(x) = x3 - 5x Write out approach, questions and areas of confusion

Non-Routine Goal #2 Problem LT 2B • If f(x) is an even function, determine if g(x) is even, odd or neither. Explain. • g(x) = – f(x) • g(x) = f(– x) • g(x) = f(x) – 2 • g(x) = – f(x – 2)

Compare to Solutions Annotate your “work” by completing the following (use a different color) • Answer questions • Isolate and resolve areas of confusion (without copying how to do the problem) • Identify areas of strength • Analyze reasoning (Approach, Plan, Execution) • Analyze written response

Based on Goal Problem #2, Current Level of Mastery? 4: Demonstrate recall, solve routine problems, and solve non-routine problems (levels 1, 2, 3). Teach concept to a peer 3: Demonstrate recall, solve routine, approach and create plans that might work for non-routine problems (levels 1, 2, and part of 3). Explain how I reasoned through a routine problem 2: Demonstrate recall and solve routine problems (levels 1 & 2). Explain how to approach and solve a problem 1: Demonstrate recall (level 1). I do not know how to use the concepts that I know to create a plan that works NY: Not sure what facts apply and cannot remember how to approach any of the problems. I really don’t get it yet, but I am trying different things I know.

Explain the relationship between these three pictures

Explain the relationship

Goal Problems #1 (LT 2C Translations) Recall & Reproduction Sketch the graph of – f(x + 3) – 1. Routine Compare each function to the graph of f(x) = |x|. y = |x| – 3 y = |–x| y = |0.5 x| Write out approach, questions and areas of confusion

Active Practice: Learning Target 2C Translations Note to Self in Margin of notes • My Personal Learning Goal during active practice. • My choices to ensure action levels up my learning today. • How will I know if my choices are working? When will I try the Non-Routine? When will I build my concept map?

Non-Routine Problem LT 2B Translations • Determine if the following are true or false. Justify your claim. • The graph of y = f(–x) is a reflection of the graph of y = f(x) over the x-axis. • The graph of y = –f(x) is a reflection of the graph of y = f(x) over the y-axis. • The graph of y = –f(x) is a reflection of the graph of y = f(–x) over the x-axis.

Recall & Reproduction Goal Problems #2 LT 2C Translations Routine Piecewise, use quadratic, exp, logs, abs value. Step fcn Explain the translations necessary to turn the graph of f(x) into the graph of g(x). f(x) = need a log function with all shifts g(x) = need a e to the x with all shifts Write out approach, questions and areas of confusion

Non-Routine Goal #2 Problem LT 2B • The graph of f(x) has x-intercepts at x = –1 and x = 4. Use this to find the x-intercepts of the given graph. If not possible, explain why. • f(–x) • f(x – 2) • 2f(x) • f(x) – 1

Non-Routine Goal #3 Problem LT 2B

Next Steps • GRASP • Build Concept Map • Strengthen Concept Map • Complete individual active sensemaking practice problems

Build Concept Map Requires doing more problems Use Learning Targets & Notes, Goal Problems, Quick Check, and problems from the book to design concepts and categories What is in black, includes visuals (labeled diagrams, types of problems) How to Approach is green Focus on connections where you explain how to think about using information about the concepts to figure out non-routine problems Plans to use the information and why this will lead to an accurate solution Questions I ask myself as I think through my approach, plans, and execution

Strengthen Concept Map Based on Qualitative & Quantitative Evidence, add to your concept map • Concepts, Definitions, Visuals, Explanation of “what” • Approach for concepts • Reasons the concepts are useful in creating plans (connections) • Questions I pose to myself as I move through my concept map

Strengthen Concept Map with Problems From Learning Targets 2A, 2B, 2C & Goal Problems Non-Routine, Routine, Recall & Reproduction Select from learning targets…..which ones have I not done yet? Choose one of each and write out myapproach, plans if needed, and questions I pose to myself as I move through my solution pathway

LT 2E: Describe the limiting behavior Approach Independently Create 2 Plans that might work by communicating with partner Approach Create Plans

Relationships between Inputs & Outputs