MODULE FOUR. This module addresses functions. SC Academic Elementary Algebra Standards:

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1 MODULE FOUR This mdule addresses functins SC Academic Standards: EA-3.1 Classify a relatinship as being either a functin r nt a functin when given data as a table, set f rdered pairs, r graph. EA-3.2 Use functin ntatin t represent functinal relatinships. EA-3.3 Carry ut a prcedure t evaluate a functin fr a given element in the dmain. EA-3.4 Analyze the graph f a cntinuus functin t determine the dmain and range f the functin. EA-3.5 Carry ut a prcedure t graph parent functins 2 1 (including y x, y x, y x, y x, and y ). x The resurces prvided in this mdule are nt all inclusive. They are prvided t begin t build the cnceptual fundatin students need. Additinal resurces will be required t develp the cncepts. 1

2 Lessn # 1 Tpic: Classifying a relatin as a functin r nt a functin Standard (s): EA 3.1 I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills needed t meet this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade, students translate amng verbal, graphic, tabular, and algebraic representatins f linear functins (8-3.1). They als classify relatinships between tw variables in graphs, tables, and/r equatins as either linear r nnlinear (8-3.5). In, students classify a relatinship as being either a functin r nt a functin when given data as a table, set f rdered pairs, r graph (EA-3.1). In Intermediate Algebra, students carry ut prcedures t perfrm peratins n plynmial functins (including f(x) + g(x), f(x) g(x), f(x) g(x), and f(x)/g(x)) (IA-2.5). They als apply a prcedure t write the equatin f a cmpsitin f given functins (IA-2.6). In additin, students carry ut a prcedure t graph discntinuus functins (including piecewise and step functins) (IA-2.9) and determine the dmain and range f discntinuus functins (including piecewise and step functins) (IA-2.10). Taxnmy Level Cgnitive Prcess Dimensin: Understand Knwledge Dimensin: Cnceptual Knwledge Key Cncepts Functin Relatinship II. Teaching the Lessn In this lessn, students classify relatins as either a functin r nt a functin. This can seem t e a very trivial tpic t students because they may nt see the applicatin f the cncept in the real wrld. T illustrate the real wrld cnnectin, present tw input/utput machines: an ATM machine and accunts. With an ATM machine, the input frm yur card is yur accunt number and the utput is yur mney. S the rdered pair is (yur accunt number, yur mney). Create sample rdered pairs using yur student s name. Fr example, (Cindy s acct #, Cindy s $), (Jack s acct #, Jack s $), etc Then include the rdered pair (Cindy s acct #, Sandra s $) 2

3 and ask wuld that be OK? Well it might be gd fr Cindy but nt fr Sandra. S we have ne input with tw utput and that isn t gd because banks wuld be ut f business if they culd nt guarantee a functinal relatinship ne input paired with ne utput. G thrugh the same prcess with the accunts example where the input is yur acct identifier (user name/passwrd) and the utput is yur s. The rder pair is (yur acct identifier, yur s). Then ask students t cme up with ther real wrld situatins where are functinal relatinship is imprtant. Sme examples are car keys, phne numbers, huse keys, scial security numbers. Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Use the definitin f a functin t classify a set f rdered pairs in a table r list as a functin r nt a functin. Use the definitin f a functin t classify a set f rdered pairs in a mapping as a functin r nt a functin. Use the vertical line test and the graph f a cntinuus relatinship t classify the relatinship as a functin r nt a functin. Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. Classify the relatinship shwn in the table belw as a functin r nt a functin. X Y Answer: nt a functin Classify the relatinship shwn in the table belw as a functin r nt a functin. X Y Answer: functin 3

4 Classify the relatinship shwn in the table belw as a functin r nt a functin. X Y Answer: functin Classify the relatinships shwn belw a functin r nt a functin. {(0,2), (3,7), (4,10), (6,15), (8,10)} Answer: functin {(0,6), (3,6), (4,6), (6,6), (8,6)} Answer: functin {(5,2), (5,7), (5,10), (5,15), (5,10)} Answer: nt a functin Classify the relatinship shwn n the graph belw as a functin r nt a functin. Answer: functin Classify the relatinship shwn n the graph belw as a functin r nt a functin. Answer: nt a functin 4

5 Classify the relatinship shwn n the mapping belw as a functin r nt a functin Answer: functin Classify the relatinship shwn n the mapping belw as a functin r nt a functin Answer: nt a Functin Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within : Classify piecewise relatinships as a functin r nt a functin. Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within. 5

6 Classify the relatinship shwn n the graph belw as a functin r nt a functin. III. Miscnceptins/Cmmn Errrs Students may nt understand that tw rdered pairs with the same y- crdinate but different x-crdinates are pssible rdered pairs fr a functin. Students may cnfuse discrete and cntinuus data. Tables, lists, and mappings represent discrete data, unless the infrmatin prvided with the table indicates that the given pints are pints n a cntinuus graph. When given a graph, students differentiate between discrete and cntinuus data mre easily. Technlgy Nte Use technlgy where apprpriate. Assessing the Lessn The bjective f this indicatr is fr the student t understand the definitin f a functin s that the student can classify a relatinship as being either a functin r nt a functin when given data as a table, set f rdered pairs, r graph. In additin t classifying relatinships as functins r nt functins, students shuld be able t exemplify, cmpare, r explain relatinships that are functins r nt functins. Assessment Item Examples Which equatin represents the functin in the table? Input X Output y A. y=x+5 B. y=2x+3 C. y=x 2 +3 D. y=x E. y=3x+3 6

7 Lessn # 2 Tpic: Writing equatins using functin ntatin Standard (s): EA 3.2 I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills needed t meet this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade, students translate amng verbal, graphic, tabular, and algebraic representatins f linear functins (8-3.1). They als classify relatinships between tw variables in graphs, tables, and/r equatins as either linear r nnlinear (8-3.5). In, students will use functin ntatin t represent functinal relatinships (EA-3.2). In Intermediate Algebra, students carry ut prcedures t perfrm peratins n plynmial functins (including f(x) + g(x), f(x) g(x), f(x) g(x), and f(x)/g(x)) (IA-2.5). Taxnmy Level Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Cnceptual Knwledge Key Cncepts Functin Ntatin II. Teaching the Lessn In this lessn, students write relatinship functins using functin ntatin. The indicatr fr this lessn ges beynd applying the prcedure f simply replacing y with f(x). The indicatr asks students t apply cnceptual knwledge; therefre, students need t understand the relatinship between independent and dependent variable (input and utput). The next lessn n EA 3.2 asks students t evaluate functins. Yu may want t integrate that lessn here r teach it as a separate lessn. Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Use functin ntatin t represent functins given as an equatin, graph, r described in wrds. 7

8 Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. Use functin ntatin t represent the functin shwn in the graph belw. Y = x Answer: f(x) = x Use functin ntatin t represent the functin described belw. The temperature f an bject n the Celsius scale is 5/9 the difference between the Fahrenheit temperature (t) and 32 F. Answer: F(t) = 5/9(t-32) Given y = 1/x. Express this equatin in functin ntatin. Answer: f(x) = 1/x Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within : Derive the functinal relatinship. Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within Use functin ntatin t represent the linear functin shwn in the table belw. Y r X f(x) Answer: f(x) = 2x + 4 8

9 Use functin ntatin t represent the linear functinal relatinship represented belw cntaining the set f rdered pairs (x, y) r (x, f(x)). {(0, 0), (3,-9), (4,-12), (-6,18), (-8,24)} Answer: f(x) = -3x Miscnceptins/Cmmn Errrs Students may nt understand that functin ntatin defines a functin in terms f an independent variable, but des nt specify the symbl used t represent the dependent variable. Technlgy Nte When graphing a functin that is defined in functin ntatin using technlgy, y = is used even thugh the dependent variable may nt be y. III. Assessing the Lessn Assessment Guidelines: The bjective f this indicatr is fr the student t apply cnceptual knwledge t use functin ntatin t represent functinal relatinships. Assessment Item Example Which f the fllwing expresses the relatinship described belw in functin ntatin? q is a functin f r and 2 times r equals q A. q(r) = 2q B. q(r) = 2r C. r(q) = 2q D. r(q) = 2r 9

10 Lessn # 3 Tpic: Evaluating a functin given a value Standard (s): EA 3.3 I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills needed t meet this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade, students apply frmulas t determine the exact (pi) circumference and area f a circle (8-5.4) and apply frmulas t determine the perimeters and areas f trapezids (8-5.5). T determine these measurements, students substitute values fr variables in apprpriate frmulas. In, students carry ut a prcedure t evaluate a functin fr a given element in the dmain (EA-3.3).. In Precalculus, students apply a prcedure t evaluate trignmetric expressins. (PC-5.6). Thugh nt specifically stated as an indicatr, this essential skill is used in all cre areas. Taxnmy Level Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Prcedural Knwledge Key Cncepts Value f a Functin Dmain Value Range Value Substitutin Evaluatin II. Teaching the Lessn In this lessn, students evaluate a functin fr a given value. Students evaluated expressins in Mdule 1 and are nw applying the prcedure t expressins written in functin ntatin. This lessn may be integrated int the previus lessn where students are representing relatinships using functin ntatin. Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Substitute a given value fr the independent variable and use the rder f peratins t evaluate the functin fr the given element. 10

11 Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. Find the value f the functin f(x) = 4x + 6 fr x = -2. Given g(t) = 2t 6, evaluate g(2). Fr y = x + 2, what is the value f y when x = -4? Find the value f y fr x = ½, if y = 1/x. Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within : Evaluate a functin fr elements in the dmain with variable values. Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within. Evaluate g(t) = 2t - 6 fr t = a. Miscnceptins/Cmmn Errrs Students may think that fr g(t) = 2t 6 that g(t) means g multiplied by t. Technlgy Nte Use technlgy where apprpriate. III. Assessing the Lessn Assessment Guidelines: The bjective f this indicatr is fr the student t carry ut a prcedure t evaluate a functin fr a given element in the dmain. Assessment Item Example See examples f essential tasks 11

12 Lessn # 4 Tpic: Analyze the graph t determine the dmain and range Standard (s): EA 3.4 I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills needed t meet this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade, students use rdered pairs, equatins, intercepts, and intersectins t lcate pints and lines in a crdinate plane (8-4.2). In, students analyze the graph f a cntinuus functin t determine the dmain and range f the functin (EA-3.4). In Intermediate Algebra, students carry ut a prcedure t determine the dmain and range f discntinuus functins (including piecewise and step functins) (IA-2.10). Taxnmy Level Cgnitive Prcess Dimensin: Analyze Knwledge Dimensin: Prcedural Knwledge Key Cncepts Dmain Range II. Teaching the Lessn In this lessn, students analyze cntinuus graphs t determine the dmain and range. Students may nt have prir knwledge with this tpic; therefre this lessn may need t be revisited thrughut the curse in rder t ensure student prficiency. Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Determine the set f x-crdinates f pints n the graph. Describe the set f x-crdinates using wrds r mathematical expressins t specify the dmain f a cntinuus functin. Determine the set f y-crdinates f pints n the graph. Describe the set f y-crdinates using wrds r mathematical expressins t specify the range f a cntinuus functin. 12

13 Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. Analyze the graph belw t determine the dmain and range f the functin. Arrws are n the ends. Answer: Dmain: all real numbers Range: y 3 Analyze the graph belw t determine the dmain and range f the functin. Arrws are n the ends. Answer: Dmain: all real numbers Range: all real numbers Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within : Analyze the graph f nn-cntinuus functins that are piecewise, including step functins. 13

14 Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within. This piecewise functin is nt cntinuus, s the indicatr des nt apply. The step functin (y = [x]) shwn at the left is nt cntinuus, s the indicatr des nt apply. The greatest integer functin is nt cntinuus. Miscnceptins/Cmmn Errrs Students smetimes reverse the dmain and range f functins. Technlgy Nte Students may enter a functin, lk a graph, and use the trace feature. Students may use the table feature. 14

15 III. Assessing the Lessn Assessment Guidelines: The bjective f this indicatr is fr the student t analyze the graph f a cntinuus functin t determine the dmain and range f the functin. Assessment Item Example Analyze the graph belw t determine the dmain and range f the functin. 15

16 Lessn # 5 Tpic: Graphing parent functins Standard (s): EA 3.5 I. Planning the Lessn The first bullet under the Cntinuum f Knwledge represents student s prir knwledge and/r skills needed t meet this standard. It is recmmended that students are pre-assessed n this prir knwledge. Cntinuum f Knwledge In 8 th grade, students use rdered pairs, equatins, intercepts, and intersectins t lcate pints and lines in a crdinate plane (8-4.2). In, students carry ut a prcedure t graph parent 2 1 functins (including y x, y x, y x, y x, and y ) (EA-3.5). x In Intermediate Algebra, students carry ut a prcedure t graph translatins f parent functins 2 1 (including y x, y x, y x, y x, and y ) (IA-2.7). x Taxnmy Level Cgnitive Prcess Dimensin: Apply Knwledge Dimensin: Prcedural Knwledge Key Cncepts x-crdinate y-crdinate Ordered pair Crdinate plane II. Teaching the Lessn In this lessn, students are intrduced t the parent functins. Student s prir knwledge may nt include functins ther than linear functins. This lessn may be integrated int ther lessn later in the curse r it may be addressed in its entirety here. Students explred the graphs f linear functins in Mdule Three and inverse relatinships in Mdule Tw. They will explre quadratic functins in Mdule 8. The abslute value and square functins are nt addressed in ther mdules and may be addressed separately r integrated int ther mdules. Essential Learning and Understanding It is essential fr students t d the fllwing fr the attainment f this indicatr: Use the equatin f a parent functin t determine a set f rdered pairs that lie n the graph f the equatin, lcate thse 16

17 pints in a crdinate plane, and sketch a cntinuus graph by analyzing the pattern f the pints pltted. Examples f Essential Tasks These examples f essential tasks are nt all inclusive. They are prvided t give additinal clarificatin f pssible tasks that students shuld be able t successfully cmplete. Sketch a graph f y = x in a crdinate plane. Sketch a graph f y = x² in a crdinate plane. Sketch a graph f y = x in a crdinate plane. Sketch a graph f y = x in a crdinate plane. 1 Sketch a graph f y = in a crdinate plane. x Nn-Essential Learning and Understanding It is nt essential fr students t d the fllwing fr the attainment f this indicatr but culd be imprtant fr the attainment f ther indicatrs within : Graph transfrmatins f parent functins Examples f Nn-Essential Tasks The examples f nn-essential tasks given belw are nt essential fr the attainment f this particular indicatr but culd be imprtant fr the attainment f ther indicatrs within. This is a translatin f y = x² (y = x² + 2). It is nt essential fr students t graph translatins f parent functins. Miscnceptins/Cmmn Errrs Nne nted Technlgy Nte If students graph y = 1/x in cnnected mde, then the calculatr will incrrectly cnnect the parts f the graph in quadrant I and quadrant III. A mre accurate representatin is prduced when students graph in the uncnnected mde. 17

18 III. Assessing the Lessn Assessment Guidelines: The bjective f this indicatr is fr the student t carry ut a prcedure t graph parent functins 2 1 (including y x, y x, y x, y x, and y ). x 18