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Algebra 2 andTrigonometryChapter 4:FUNCTIONSName:Teacher:Pd:

Table of ContentsDay1: Chapter 4-1: Functions; Domain and RangeSWBAT: Identify the domain and range of relations and functionsPgs. #1 - 5Hw: pg 126 in textbook. #1 - 11Pg.133 in textbook #3 – 12Day2: Chapter 4-2: Function NotationSWBAT: Evaluate FunctionsPgs. #6 - 10HW: pg 129 in textbook. #3 – 15, 17Day3: Chapter 4: Functions with Restricted DomainsSWBAT: Calculate restricted domains of functionsPgs. #11 - 14Hw: Worksheet in Packet on Pages 15-16Day4: Chapter 4-4: Graphing Absolute Value FunctionsSWBAT: (1) Graph Absolute Value Functions(2) Translate Absolute Value FunctionsPgs. #17 – 23HW: Worksheet in Packet on Pages 24- 26Day5: Chapter 4-5/4-6: Transformations of Quadratic and Other functionsSWBAT: Transform Quadratic and Other functionsPgs. #27 – 31Hw: Worksheet in Packet on Pages 32-35Day6: Chapter 4-7: Composition of FunctionsSWBAT: Evaluate the composition of a functionPgs. #36 – 40Hw: Worksheet in Packet on Pages 41-42Day7: Chapter 4-8: Inverse FunctionsSWBAT: Find the Inverse of a FunctionPgs. #43 – 48Hw: Worksheet in Packet on Pages 49-50Day8: Chapter 4-10: Inverse VariationSWBAT: Solve Problems involving Inverse VariationPgs. #51 – 45Hw: Worksheet in Packet on Page 56HOMEWORK ANSWER KEYS – STARTS AT PAGE 57

Chapter 4–1 – Relations and Functions (Day 1)SWBAT: Identify the domain and range of relations and functionsA set of ordered pairs is called a . Ex: {() () () (} The domain of a relation is the set of all valuesThe range of a relation is the set of all values.Notation Use { } if the D/R has only a few values Use Set Notation otherwise{x -2}{y -1}1

For each relation below, state the domain and range.Example 1:Example 2:FunctionsA function is a relation where each x goes to only one y No x values are repeated among ordered pairs A graph would pass the Vertical Line Test Any vertical line only crosses graph once It is OK if the y‐values are repeated2

One-to-One FunctionsA one-to-one function (1-1) is function relation in which each member of therange also corresponds to one and only one member of the domain. No y values are repeated among ordered pairs A graph would pass the Horizontal Line TestFor each function below, determine if it is One-to-One.Example 3:Example 4:3

Am I a function? Am I One-to-One?If your answer to “Is it a function” or “is it a 1-1 function” is “no” explain why not.Domain a.b.TomLuisIrvinMarcEboneNinaRobynUnsha{(-1, 5), (2, 5), (2, 4), (-3, 1)}Range Function?1-1 Function?Domain Range Function?1-1 Function?c.Domain Range Function?1-1 Function?d.Domain Range Function?1-1 Function?e. y -(x 2)2 8Domain Range Function?1-1 Function?f. Domain Range Function?1-1 Function?4

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Chapter 4–2 – FUNCTION Notation (Day 2)SWBAT:Evaluate FunctionsWarm – Up:Determine the domain and range of the relation below.Determine if the relation is a function and if it is a one-to-onefunction.Domain Range Function?1-1 Function?Function Notation x is an independent variable Y is the dependent variable because its valuedepends on the given x‐value Y f(x)– Means y is a function of x (dependant on x)– Read “f of x”– F is the name of the function– X is the independent variable6

If you want to evaluate a function at, for example, the x-value of 3,we write “determine ( ).” Simply substitute x in the equationand evaluate:Example: If ( )( ), find ( )( )( )Example 1: If f(x) 2x 3, finda. f(-4) b. f(a 1) c. f(2x) d. f(x2) Example 2: If f(x) x2 – 6, finda. f(2)b. fnd f(n - 2)c. find f(3x)d. If the domain of f(x) x2 – 6 is {x -2 x 2}, find the range of the function.7

Example 3: The graph of function f is shown below. Find:a. f(-1)b. f(0)c. f(1)d. f(3)Practice Section: Evaluating Functions1. If ( )3. If f(x) ,find (, find f(8)5. If f(x) x2 - 4x , find f(-2)).2. If f(x) , find f(6).4. If f(x) 4x - 5 , find f(-2)6. If g(x) 3x 4 and the domain is{x -1 x 7}, find the range.8

7.Answers:a.b.c.d.e.f.g.h.i.9

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Chapter 4 – Functions with Restricted Domains(Day 3)SWBAT:Calculate restricted domains of functionsWarm – Up:Functions with Restricted DomainsAny equation that can be written as “y ” with no symbol is a function.Almost every any function we study this year has the domain “All RealNumbers” ( ) which means that you are allowed to use ANY VALUE OF Xyou want, and there will be some value of y that corresponds to it.Functions with Restricted Domains have some value(s) of x which cannotbe used, because it results in some undefined values of y.Functions that have no domain restriction:( )( )( ) 11

These are the three functions with restricted domains we will explore this year:Rational FunctionsSquare Root FunctionsA rational function is( )defined as ( ),( )A square root functionhas a square root in it!( ) ( )where ( ) and ( )are also functions of x.Example:( )try the value x 3.Rational Fractions areundefined when thedenominator 0Put the two together and you have arational function with a square root inthe denominator.Example: ( ) try the value x -10.Set the den. 0 andSolve. These are therestricted values.Example:( ) try the value x -4.Square Root functions areundefined when theradicand is 0( )( )To restrict the domain:Combination of the two (acomposition of a rationalfunction and a square rootfunction)These functions are undefinedwhen the radicand in thedenominator 0 To restrict the domain:Set the radicand 0.Solve. These are therestricted values.f ( x) 34x 8To restrict the domain:Set the denominator’sradicand 0. Solve. Theseare the restricted values.Determine the domain of each of the following rational functions:a) ( )b) ( )c) ( )12

Determine the domain of each of the following rational functions:e) ( )d) ( )f)( )Determine the domain of each of the following square root function:g) f ( x ) xh) f ( x ) x 3i) f ( x ) 2 x 6j)f ( x) 3 x 4 8k)( ) Determine the domain of each of the following compositions of square root and rational functionsl)f ( x) 34x 8m)( ) n)( ) 13

Summary/Closure:Type of Function ExampleDomainRangeLinearQuadraticAbsolute ValueSquare RootRationalFunctionComposition ofRationalFunction andSquare RootFunctionDo Not DetermineDo Not DetermineExit Ticket:14

Restricted Domain Homework – Day 3Work On Problems #5, 8, 10, 11, 13, 15 ,16-24 ,26, 28, 3315

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GRAPHING ABSOLUTE VALUE FUNCTIONS (Day 4)SWBAT: Graph Absolute Value functionsWarm – Up:In Unit 2, we talked about absolute value in terms of the distance a number is away from zero on a number line.We will now investigate the graph of the absolute value function.Graph: VERTEXThe vertex of the absolute value function is the point where the function changes direction.Which coordinates are the turning points (vertex) of the graph above? ( , )17

Exercise #2: Graph the following functions using a graphing calculator.y y SUMMARYWhen y the graph is shiftedWhen y the graph is shifted18

Exercise #3: Consider the function: y and y (a) Using your graphing calculator to generate an xy-table, graph this function on thegiven grid.y y SUMMARYCompare the graphs for problem 4. Make a conjecture about functions that come in the form:y .When y When y the graph is shiftedthe graph is shifted19

A translation is a shift of a graph vertically, horizontally, or both. The resulting graph isthe same size and shape as the original but is in a different position in the plane.Practice: Writing equations of an absolute value finction from its graph.Write an equation for each translation of y shown below.a)b)c)d)e)f)20

Vertical and Horizontal TranslationsIdentify the vertex and graph each.g) y h) yVertex: ( , )i)y - 2Vertex: ( , ) Vertex: ( , )j) y 3Vertex: ( , )21

Lastly, let’s observe what happens when the coefficient changes in front ofthe absolute value of x.y y Compare the graphs from above. Make a conjecture about functions that come in the form:y When y the graph opensWhen y the graph is opens(which means its reflected over the axis)y y Compare the graphs from above. Make a conjecture about functions that come in the form:y When yWhen the graphis than the graphis thany .y .22

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Homework – Writing and Graphing Functions – Day 44.24

In examples 5 – 13, write an equation for each translation of y 5.6.7.8.9.10.9 units up6 units downright 4 unitsleft 12 units8 units up, 10 units left3 units down, 5 units right 11.12.13.25

In examples 14 and 15, identify the vertex and graph each.14) y Vertex: ( , )16) On the set of axes below, graph and label theequations y y 2 .Explain how changing the coefficient of theabsolute value from 1 to 2 affects the graph.15) y - 7Vertex: ( , )17) On the set of axes below, graph and label theequations y y .Explain how changing the coefficient of theabsolute value from 1 to affects the graph26

Chapter 4–5/4-6 –TRANSFORMATION OF QUADRATICSAND OTHER FUNCTIONS (Day 5)SWBAT: Transform Quadratic and Other functionsWarm – Up:1.2. The vertex of y (1)(2)(3)(4) - 3 is(2, 3)(-2, 3)(-2, -3)(2, -3)27

Quadratic FunctionsA quadratic function is an equation in the form y ax2 bx c, where a, b, and c are real numbers anda0. The shape of a quadratic function is a , a smooth and symmetric U-shape.Example 4: Use the table of values below to graph the quadratic of basicfunctionVertexFormQuadraticAbsolute ValueCubicSquare Root ()Vertex: (h, k) Vertex: (h, k) ()Vertex: (h, k) Vertex: (h, k)28

Knowing how a function can be transformed makes it easier to graphthe function. There are three types of transformations that can be doneto a function.Functionf(x) cf(x) - cf(x c)f(x - c)f(-x)-f(x)( )a f(x)( )f(ax)TransformationDescribe the transformation for each function.FunctionTranformation ()2 7 29

Graphing Section: Graph each.1. y - 3. y (x 2)2 12. y 4. y 2x2 - 430

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Day 5 – HWTransformation of Functionsa)( – )( )b)c)()32

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Day 6: Chapter 4–7/Composition of FunctionsSWBAT: Add, Subtract, Multiply and Divide FunctionsWarm – Up:Operations on Functions Given 2 functions f and g:NotationMeans()( )( – )( )( )( )( )( )Domain of Functions First find domain of f and g The domain of , -, or * is the intersectionof the domains of f and g The domain of must exclude values thatmake denominator 0.36

Find:a) ()( )b) ()( )c) ()( )d) ( ) ( )1)Composition of FunctionsA composition of two or more functions is when a function is performed on the result of anotherevaluated function.)( ) both are readThere are two ways to represent the composition of a function: ( ( )) or (“f of g of x”, and mean that you are taking the result of g(x) and applying that result to the function f.Example: f(x) x 2 and g(x) 5x – 4. Find f(g(3)).F(g(3)) means determine g(3), then find f(result).gf3 5(3) – 4 11 (11) 2 13(f g)(3)So, f(g(3)) 13. This is also (f g)(3).Composition of functions is generally not commutative, meaning the order matters!37

Given: f(x) 3x – 2Determine:1. (g f)(-3)4. (h g)(-6)g(x) x2 1h(x) x 3j(x) (x 2)22. (f g)(-3)3. (g h)(4)5. (f h g)(2)6. (g h f)(1)Finding a general formula for f(g(x)) or g(f(x)) requires you to go the same thing without actualvalues of x.Example: f(x) x 2 and g(x) 5x – 4. Find f(g(x)) and g(f(x)).f(g(x)) means do g(x) first, then do f(result). f( g(x) ) f(5x – 4) (5x – 4) 2 5x – 4 2 5x – 2.So, f(g(x)) 5x – 2 g( f(x) ) g( x 2 ) 5(x 2) – 4 5x 10 – 4So, g(f(x)) 5x 638

Given: f(x) 3x – 2g(x) x2 1Determine the general formulas for:7. (f g)(x)8. (g f)(x)10. (h g)(x)11. (f h)(x)h(x) x 3j(x) (x 2)29. (g h)(x)12. (h f)(x)39

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Day 6: HWWork on Problems #2-42 Every Other Even41

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