Absolute Value Functions - MR. JONES
Lesson 4-7Absolute Value FunctionsExplore Parameters of an Absolute Value Function Online Activity Use graphing technology to complete the Explore. INQUIRY How does performing an operationToday’s Goals on an absolute value function change itsgraph? Learn Graphing Absolute Value FunctionsThe absolute value function is a type of piecewise-linear function.An absolute value function is written as f(x) a x - h k, where a, h,and k are constants and f(x) 0 for all values of x.The vertex is either the lowest point or the highest point of a function.For the parent function, y x , the vertex is at the origin.Key Concept Absolute Value FunctionParent Functionx if x 0f(x) x , defined as f(x) { - x if x 0Type of GraphV-ShapedDomain:all real numbersRange:all nonnegative real numbersKey Concept Vertical Translations of Absolute Value FunctionsCopyright McGraw-Hill EducationApply translations toabsolute value functions.Apply dilations toabsolute value functions.Apply reflections toabsolute value functions.Interpret constantswithin equations ofabsolute value functions.Today’s Vocabularyabsolute value functionvertexThink About It!Why does adding apositive value of k shiftthe graph k units up?Sample answer: Thisadds k to every value ofthe output of theabsolute value. Sincethe output correspondsto the y-values, thegraph moves up k units.Learn Translations of Absolute Value FunctionsIf k 0, the graph of f(x) x is translated k units up.If k 0, the graph of f(x) x is translated k units down.Go OnlineYou can watch a videoto see how to describetranslations of functions.Key Concept Horizontal Translations of Linear FunctionsIf h 0, the graph of f(x) x is translated h units right.If h 0, the graph of f(x) x is translated h units left.Example 1 Vertical Translations of AbsoluteValue FunctionsyDescribe the translation ing(x) x - 3 as it relates to thegraph of the parent function.Graph the parent function, f(x) x , forabsolute values.Graph absolute valuefunctions.Of(x)xg(x)The constant, k, is outside the absolutevalue signs, so k affects the y-values. Thegraph will be a vertical translation.(continued on the next page)Study TipHorizontal ShiftsRemember that thegeneral form of anabsolute value functionis y a x - h k. So,y x 7 is actuallyy x ( 7) in thefunction’s general form.Lesson 4-7 Absolute Value Functions267THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.
Your NotesSince f(x) x , g(x) f(x) k where k -3.g(x) x - 3 g(x) f(x) (-3)The value of k is less than 0, so the graph will be translated k unitsdown, or 3 units down.g(x) x - 3 is a translation of the graph of the parent function3 units down.Example 2 Horizontal Translations of AbsoluteValue FunctionsDescribe the translation inj(x) x - 4 as it relates to theparent function.yGraph the parent function, f(x) x ,for absolute values.The constant, h, is inside the absolute valuesigns, so h affects the input or, x-values.The graph will be a horizontal translation.j(x)f(x)OxSince f(x) x , j(x) f(x - h), where h 4.j(x) x - 4 j(x) f(x - 4)The value of h is greater than 0, so the graph will be translated h unitsright, or 4 units right.Think About It!Since the vertex of theparent function is atthe origin, what is aquick way to determinewhere the vertex is ofq(x) x h k?Emilio says that thegraph of g(x) x 1 1is the same graph asf(x) x . Is he correct?Why or why not?Sample answer: No; thegraph of g(x) x 1 1is a graph that has beentranslated 1 unit left and1 unit down from theparent function, f(x) x .Example 3 Multiple Translations of AbsoluteValue FunctionsDescribe the translation in g(x) x - 2 3as it relates to the graph of the parentfunction.yg(x)The equation has both h and k values. Theinput and output will be affected by theconstants. The graph of f(x) x is verticallyand horizontally translated.f(x)OSince f(x) x , g(x) f(x - h) k where h 2 and k 3.xCopyright McGraw-Hill EducationSample answer: Thevertex will be at (h, k).j(x) x - 4 is the translation of the graph of the parent function4 units right.Because h 2 and k 3 , the graph is translated 2 units rightup .and 3 unitsg(x) x - 2 3 is the translation of the graph of the parent function2 units right and 3 units up .Go Online You can complete an Extra Example online.268 Module 4 Linear and Nonlinear FunctionsTHIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.
Example 4 Identify Absolute Value Functionsfrom GraphsUse the graph of the function to write itsequation.The graph is the translation of the parent graph1 unit to the right.yf(x)g(x)OxGeneral equation for ag(x) x - h horizontal translationg(x) x - 1 The vertex is 1 unit to theright of the origin.Example 5 Identify Absolute Value Functionsfrom Graphs (Multiple Translations)Use the graph of the function to write itsequation.The graph is a translation of the parent graph2 units to the left and 5 units down.yf(x)OGeneral equation for ag(x) x - h k xg(x)translationsg(x) x - (-2) k The vertex is 2 units leftof the origin.g(x) x - (-2) (-5) The vertex is 5 units down of the origin.Copyright McGraw-Hill Educationg(x) x 2 - 5Simplify.Learn Dilations of Absolute Value FunctionsMultiplying by a constant a after evaluating an absolute value functioncreates a vertical change, either a stretch or compression.Key Concept Vertical Dilations of Absolute Value FunctionsIf a 1, the graph of f(x) x is stretched vertically.If 0 a 1, the graph of f(x) x is compressed vertically.When an input is multiplied by a constant a before for the absolutevalue is evaluated, a horizontal change occurs.Key Concept Vertical Dilations of Absolute Value FunctionsIf a 1, the graph of f(x) x is compressed horizontally.If 0 a 1, the graph of f(x) x is stretched horizontally.Talk About It!How is the valueof a in an absolutevalue function relatedto slope? Explain.Sample answer: Thevalue of a determinesthe slope of each partof the graph. Thefunction y a x canalso be written asax it x 0 f(x) { -ax if x 0where a and -a arethe slopes of the rays.Go Online You can complete an Extra Example online.Lesson 4-7 Absolute Value Functions269THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.
Example 6 Dilations of the Form a x When a 15Describe the dilation in g(x) 2 x as it relates to the graph of theparent function.5 2 .Since f(x) x , g(x) a f(x), where a 5Go OnlineYou can watch a videoto see how to describedilations of functions.5g(x) 2 x is a vertical stretch of the graph of the parent graph.x x -4 -4 4 -2 2 0 0 2 2 4 4-2Think About It!How are a x and ax evaluated differently?Sample answer: In a x ,the absolute value ofthe input is evaluatedbefore multiplyingby a. In ax , the input isfirst multiplied by a andthen the absolute valueis evaluated.0245 2 x y(x, g(x))10(-4, 10)5(-2, 5)0(0 ibe the reflection in j(x) x 3 5 as it relates to thegraph of the parent function. x 3 x- x 3 - x 3 5(x, j(x))-2-2 5 3(-5, 3)-1-1 5 4(-4, 4) 3 -2 2-5 -5-4 -4 3 -1 1-3 -3 3 0 000 5 5(-3, 5)-2 -2 3 1 1-1-1 5 4(-2, 4)-1 -1 3 2 2-2-2 5 3(-1, 3)First, the absolute value of x 3 is evaluated.Then, the function is multiplied by a. Finally, 5is added to the function.Plot the points from the table.Since f(x) x , j(x) 1 · a · f(x)where a 1, h 3, and k 5 .j(x) - x 3 5 j(x) -1 · f(x 3) 58642 8 6 4 2 O 4 6 8yf(x)2 4 6 8xj(x)j(x) - x 3 5 is the graph of the parent function reflected acrossthe x-axis , and translated 3 units left and 5 units up .Example 10 Graphs of y a x 3Describe the reflection in q(x) - 4 x as it relates to the graph ofthe parent function.Plot the points from thetable.x x -8 -8 8-4 -4 0484 0 0 4 4 8 83(x, q(x))-6(-8, -6)-3(-4, -3)0(0, 0)3(4, 3)6(8, 6)- 4 x Copyright McGraw-Hill EducationFirst, the absolute value ofx is evaluated. Then, thefunction is multiplied by a.Finally, 5 is added to thefunction.Since f(x) x , q(x) -1 · a · f(x)where3a - 4 .q(x) - x 3 5 3q(x) - 4 f (x)3q(x) - 4 x is the graph of the parent functionreflected across the x-axis , andvertically compressed .yf(x)Oxq(x)Go Online You can complete an Extra Example online.272 Module 4 Linear and Nonlinear FunctionsTHIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.
Example 11 Graphs of y -ax Describe the reflection in g(x) 4x as it relates to the graph ofthe parent function.Think About It!Describe how thegraph of y -ax isrelated to the parentfunction.First, the input is multiplied by a. Then the absolute value of ax isevaluated.ySince f(x) x , g(x) f( 1 · a · x)g(x)where a 4 .g(x) -4x g(x) f( 1 · 4 · x)f(x)g(x) -4x is the graph of the parentfunction reflected across the y-axis andhorizontally compressed .OxLearn Transformations of Absolute Value FunctionsYou can use the equation of a function to understand the behavior ofthe function. Since the constants a, h, and k affect the function indifferent ways, they can help develop an accurate graph of thefunction.Sample answer: Sincex is multiplied by -abefore the absolutevalue is evaluated,the graph would bestretched orcompressed,depending on thevalue of a, but notreflected across thex-axis.Concept Summary Transformations of Graphs of Absolute Value Functionsg(x) a x - h kHorizontal Translation, hVertical Translation, kIf h 0 , the graph of f(x) x istranslated h units right .If k 0 , the graph of f(x) x istranslated k units up .If h 0, the graph of f(x) x istranslated h units left.If k 0, the graph of f(x) x istranslated k units down.yyh 0k 0h 0k 0Copyright McGraw-Hill EducationOxReflection, aOSample answer: Thegraph is being reflectedacross the y-axis as aresult of multiplying x by-a. There is a reflectionoccurring, but the finalgraph appears the sameas y ax . Dilation, aIf a 0, the graph opens up.If a 0 , the graph opens down.yIf a 1 , the graph of f(x) x isstretched vertically.If 0 a 1, the graph iscompressed vertically.ya 0OxWhy does there appearto be no reflection forthe graph of y ax ?a 1xa 00 a 1OGo Online You can complete an Extra Example online.xLesson 4-7 Absolute Value Functions273THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.
Choose the phrase that best describes how each parameter affectsthe graph of g(x) -5 x - 2 3 in relation to the parent function.-5Reflects and stretchesvertically2Translates right3Translates upTranslates rightTranslates leftTranslates upTranslates downStretchesReflects andvertically onlycompresses verticallyCompressesReflects and stretchesvertically onlyverticallyExample 12 Graph an Absolute Value Function withMultiple TranslationsWatch Out!Dilations andTranslationsDon’t assume thatj(x) 2 x 5 1 andp(x) 2x - 5 1 arethe same graph.Functions are evaluateddifferently dependingon whether a is insideor outside the absolutevalue symbols. It mightbe best to create atable to generate anaccurate graph.a 1The graph is notreflected or dilatedin relation to theparent function.h -1The graph istranslated 1 unit leftfrom the parentfunction.k -4The graph istranslated 4units down from theparent function.yOxyOThe graph of g(x) x 1 - 4 is the graph ofthe parent function translated 1 unit left and4 units down without dilation or reflection.The domain is all real numbers . The range isall real numbers greater than or equal to -4 .xyf(x)Oxg(x)Copyright McGraw-Hill EducationGo OnlineYou can watch a videoto see how to graph atransformed absolutevalue function.Graph g(x) x 1 4. State the domain and range.Go Online You can complete an Extra Example online.274 Module 4 Linear and Nonlinear FunctionsTHIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.
Example 13 Graph an Absolute Value Function withTranslations and DilationGraph j(x) 3x - 6 . State the domainand range.Because a is inside the absolute valuesymbols, the effect of h on the translationchanges.yj(x)f(x)OxEvaluate the function for several values ofx to find points on the graph.x(x, j(x))0(0, 6)1(1, 3)2(2, 0)3(3, 3)4(4, 6)The graph of j(x) 3x - 6 is the graph of the parent functioncompressed horizontally and translated 2 units right.The domain is all real numbers . The range is all real numbers greaterthan or equal to 0 .Think About It!How is the verticaltranslation k of anabsolute value functionrelated to its range?Example 14 Graph an Absolute Value Function withTranslations and ReflectionCopyright McGraw-Hill EducationGraph p(x) - x - 3 5. State thedomain and range.In p(x) - x - 3 5, the parentfunction is reflected across the x-axisbecause the absolute value is beingmultiplied by -1.The function is then translated3 units right.yf(x)Oxp(x)Finally, the function is translated 5 units up.p(x) - x - 3 5 is the graph of the parent function translated3 units right and 5 units up and reflected across the x-axis .The domain is all real numbers . The range is all real numbers lessthan or equal to 5 .Sample answer: Sincea vertical translationaffects the location ofthe minimum ormaximum point, therange will be greaterthan or equal to or lessthan or equal to thevalue of k, dependingupon whether thefunction has beenreflected.Go Online You can complete an Extra Example online.Lesson 4-7 Absolute Value Functions275THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.
Example 15 Apply Graphs of Absolute ValueFunctionsBUILDINGS Determine anyabsolute value functionthat models the shape ofThe Palace of Peace andReconciliation.Think About It!How is the verticaltranslation k of anabsolute value functionrelated to its range?Sample answer: Since avertical translation affectsthe location of theminimum or maximumpoint, the range will begreater than or equal to orless than or equal to thevalue of k, depending uponwhether the function hasbeen reflected.To write the equation forthe absolute valuefunction, we mustdetermine the valuesof a, h, and k in f(x) a x - h k from thegraph.(0, 62)( 31, 0)(31, 0)OxIf we consider the absolute value as a piecewise function, we can findthe slope of one side of the graph to determine the value of a.Because this function opens downward, the graph is a reflection of theparent graph across the x-axis. So we know that the a-value in theequation should be negative.y2 - y1m x - x 2The Slope Formula10 - 62 31 - 0 62- 31 (0, 62) (x1, y1) and (31, 0) (x2, y2)or - 2Next, notice that the vertex is not located at the origin. It has beentranslated. The absolute value function is not shifted left or right, buthas been translated 62 units up from the origin.a 2, h 0, k 2y -2 x 62Simplify.Copyright McGraw-Hill Educationy -2 x - 0 62So, y -2 x 62 models the shape of The Palace of Peace andReconciliation.1200GLASS PRODUCTION Certainy 119 x 9 1100800Dmitry Chulov/123RFGo Online topractice what you’velearned about graphingspecial functions in thePut It All Together overLessons 4–6 through4–7.types of glass heat and cool at anearly constant rate when theyare melted to create new glassproducts. Use the graph todetermine the equation thatrepresents this process.1000Temperature ( C)Check(9, 1100)6004002000(0, 29)4(18, 29)81216Time (hours)20Go Online You can complete an Extra Example online.276 Module 4 Linear and Nonlinear FunctionsTHIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.
Absolute Value Functions Lesson 4-7 Today’s Vocabulary absolute value function vertex Learn Graphing Absolute Value Functions The absolute value function is a type of piecewise-linear function. An absolute value function is written as f(x) a x-h k, where a, h, and k
1.1 Lesson 4 Chapter 1 Operations with Integers Key Vocabulary integer, p. 4 absolute value, p. 4 Absolute Value Words The absolute value of an integer is the distance between the number and 0 on a number line. The absolute value of a number a is written as a . 01 4 units 4 units 52 4 3 2 1 34 5
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The mean absolute deviation of a set of data is the average distance between each data value and the mean. 1. Find the mean. 2. Find the distance between each data value and the mean. . Sixth Grade Seventh Grade 88 116 94 108 112 124 144 91 97 122 128 132 9.The table shows the lengths of the longest bridges in the United BRIDGES States and in .File Size: 2MBPage Count: 6Explore furtherMean Absolute Deviation Worksheet 1.pdfdocs.google.comMean Absolute Deviation Worksheetswww.mathworksheets4kids.comMean Absolute Deviation Calculatorwww.alcula.comGrade 6 McGraw Hill Glencoe - Answer Keys Answer keys .www.mathpractice101.comMean Absolute Deviation Worksheet - Houston ISDwww.houstonisd.orgRecommended to you b
De nition 1.18. An absolute value kkthat satis es the stronger triangle inequality kx yk max(kxk;kyk) is called nonarchimedean (otherwise it is archimedean). The trivial absolute value is nonar-chimedean, as are all p-adic absolute values on Q. In fact, the absolute value jjis essentially the only archimedean absolute value on Q (as we shall see).
Absolute value The most important piecewise defined function in calculus is the absolute value function that is defined by x (x if x 2 (1,0]; x if x 2 [0,1). The domain of the absolute value function is R. The range of the absolute value function is the set of non-negative n
Absolute Value Functions Maximizing Algebra II Performance Student Lesson: Absolute Value Functions 361 The Fire Station Problem Answer Key A fire station is located on Main Street and has buildings at every block to the right and to the left. You will investigate the rela
Unit 5 – Introduction to Functions Module 6 – Linear Functions - Lesson 6.6 Transforming Linear Functions Assessment - Math Learning 6 Unit 6 – Piecewise-Defined Functions Module 18 – Piecewise and Absolute Value Functions - Lesson 18.1 Piecewise Functions Lesson 18.2 Absolute Va
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