Exploration of Transformations – Vertical Stretch ...



Name:Block:135658835723000Unit 3: Absolute ValueDay 1: Transformations of Absolute ValueDay 2: Characteristics of Absolute ValueDay 3: Absolute Value EquationsDay 4: Absolute Value InequalitiesAlg2 Day1: Graphing Using TRANSFORMATIONSWe will: learn a new way to graph functions I will: 4781550127000Note – graph your “original function” in colored pencilExploration of Transformations – Vertical Shifts1. Graph y = |x| on your calculator in Y1. a) Sketch a graph of the function.b) What is the vertex of the graph? ______482346026670002. Graph y = |x| + 2 on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How does the graph move? (up or down) _____c) What is the vertex of the graph? ______right46355003. Graph y = |x| - 5 on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How does the graph move? _____c) What is the vertex of the graph? ______right66040004. Graph y = |x| - 1 on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How does the graph move? _____c) What is the vertex of the graph? ______4887595178435005. Graph y = |x| + 4 on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How does the graph move? _____c) What is the vertex of the graph? ______6. Given that y = a|x – h| + k is the symbolic form of the absolute value function, what does the parameter k control? If k is positive, what direction do we move? If k is negative, what direction do we move?52641506032500Exploration of Transformations – Horizontal Shifts1. Graph y = |x| on your calculator in Y1. a) Sketch a graph of the function.b) What is the vertex of the graph? ______5269230119380002. Graph y = |x - 1| on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How does the graph move? Left or Right? _____c) What was the SIGN inside the absolute value?d) What is the vertex of the graph? ______52825657620003. Graph y = |x - 5| on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How does the graph move? Left or Right? _____c) What was the SIGN inside the absolute value?d) What is the vertex of the graph? ______5262245147320004. Graph y = |x + 3| on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How does the graph move? Left or Right? _____c) What was the SIGN inside the absolute value?d) What is the vertex of the graph? ______526924020055005. Graph y = |x + 7| on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How does the graph move? Left or Right? _____c) What was the SIGN inside the absolute value?d) What is the vertex of the graph? ______6. Given that y = a|x – h| + k is the symbolic from of the absolute value function, what does the parameter h control?When we have |x – h|, what direction does the graph move?When we have |x + h|, what direction does the graph move?How is the motion related to the sign of h?Exploration of Transformations – Vertical Stretch or Shrink52482758890xy-2-101200xy-2-10121. Graph y = |x| on your calculator in Y1. a) What direction does the graph open? _____b) What is the vertex of the graph? ______c) Fill in the table to the right. These coordinates are the basic ordered pairs of the absolute value function.5248275173990xy-2-101200xy-2-10122. Graph y = 2|x| on your calculator in Y2.a) What direction does the graph open? _____b) What is the vertex of the graph? ______c) Fill in the table. How do these y-coordinates compare with the y-coordinates in question 1? Is the graph fatter or skinnier?524827533020xy-2-101200xy-2-10123. Graph y = ? |x| on your calculator in Y2.a) What direction does the graph open? _____b) What is the vertex of the graph? ______c) Fill in the table. How do these y-coordinates compare with the y-coordinates in question 1? Is the graph fatter or skinnier?4. Graph y = -|x| on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How did the graph change? __________5. Graph y = -2|x| on your calculator in Y2.a) Sketch a graph of the function and the function in #1.b) How did the graph change? __________6. Given that y = a|x – h| + k is the symbolic from of the absolute value function, what does the parameter a control?Given the absolute value function y = a|x – h| + kIf a > 0, does the graph open up or down? _________ If a < 0, does the graph open up or down? _________If |a| > 1, does the graph have a vertical stretch or vertical shrink? ________ If 0 < |a| < 1, does the graph have a vertical stretch or vertical shrink? ________ What does the parameter k control? _________________________What does the parameter h control? _________________________What is the vertex? _________Now generalize…Fill in the table using your knowledge of transformations.FunctionDirection/Opening(up or down)VertexVertical Stretch or shrink1. y = ? |x + 4| - 9 2. y = - 2|x + 1| + 63. y = 4|x – 3| + 54. y = - ? |x – 7| + 35. y = 2|x + 4| - 1 6. y = -5|x – 8| - 5 Graph the functions #1 -3 above using your knowledge of transformations. 1. 2. 3. 4783455-6352353945-6354445-635Domain: _________ Domain: _________ Domain: ________Range: __________ Range: __________ Range: ________3314700114300xy00xy114300114300xy00xyGiven the absolute value equation graph, write the absolute value equation:1.2.914400222250017145008890004457700152400038862001524000114300178435xy00xy3314700178435xy00xy________________________________________4457700113030004229100113030004.18288001117600068580011176000________________________________________DAY2: The Absolute Value FunctionWe will: investigate characteristics of the absolute value functionI will:458216016891000Let’s take a look at y = x…What happens if we change every negative y-valueto a positive value? Does this sound familiar? What operation takes negative values and makes them positive?Introducing …….. the Absolute Value Function TURN&TALK: Why does the absolute value function look like a V instead of a U?We can analyze the parent function for special points and behavior..401701014859000 Domain:Range:Vertex:y-intercept:zeros (roots, x-intercepts, solutions):Increasing:Decreasing:End Behavior:Slope of right branch:35337752686050060515526860500 2. Domain:Domain:Range:Range:Vertex:Vertex:y-intercept:y-intercept:zeros (roots, x-intercepts, solutions):zeros (roots, x-intercepts, solutions):Increasing:Increasing:Decreasing:Decreasing:End Behavior:End Behavior:Slope of right branch:Slope of right branch:3.4.519430-3175003689985-635000Domain:Domain:Range:Range:Vertex:Vertex:y-intercept:y-intercept:zeros (roots, x-intercepts, solutions):zeros (roots, x-intercepts, solutions):Increasing:Increasing:Decreasing:Decreasing:End Behavior:End Behavior:Slope of right branch:Slope of right branch:5. 6. 353377545720006096004572000Domain:Domain:Range:Range:Vertex:Vertex:y-intercept:y-intercept:zeros (roots, x-intercepts, solutions):zeros (roots, x-intercepts, solutions):Increasing:Increasing:Decreasing:Decreasing:End Behavior:End Behavior:Slope of right branch:Slope of right branch:TURN&TALK: What patterns do you notice as you analyze this function family? 40665401524000Graph the inverse of the Absolute Value Function (start out with the original fn )Is the inverse a function?Day3: Absolute Value EquationsWe will: solve absolute value equationsI will: Absolute Value means __________________________________________________Absolute Value Equations182880013716000What it MEANS: Graph an Absolute Value Equation on a Number Line1.|x| = 4 As distance: |x - 0| = 4 “the set of points whose distance from 0 is equal to 4”Another way – these are two FUNCTIONS. Where are they EQUAL?… graph y=|x| and y=4…194310034290 2. |x| = 2 As distance: |x - 0| = 2 the set of points whose distance from ___ is equal to ___ As functions - What two functions are we looking at here? Where are they EQUAL?… graph y=|x| and y=2…18859501651003. |x - 4| = 3As distance: the set of points whose distance from ____ is equal to ___As functions - What two functions are we looking at here? Where are they EQUAL?… graph y=|x-4| and y=3…19431001422404. |x + 4| = 3 the set of points whose distance from ____ is equal to ___-1037239126900How we DO it ALGEBRAICALLY: Solve an Absolute Value Equation – Isolate the absolute value symbol on one side of the equal signBreak the equation into 2 derived equations – the positive case and the negative caseSolve both equationsCheck your solutions (WARNING: There may be extraneous solutions!) |x+3| = 8 2. |2x + 3| = 7TURN&TALK: Compare/contrast solving absolute value equations with solving linear equations.3. |3x + 1| - 5 = ?34. |2x + 12| = 4x 2 |x + 7| - 5 = 156. -2 |x + 7| - 5 = 15|4x + 5| = 2x + 4TURN&TALK: Given the equation |ax + b| = c, describe the values of c that would yield two solutions, one solution, and no solutions. Discuss this with your partner and write your answer.Review - Solving Linear Inequalities (This is a lead-in to Absolute Value inequalities .) Inequality Symbols: < ______________ , ________________ > ______________ , ________________Inequality Symbols: <, >, ,or Don’t forget switch the sign of the inequality when multiplying or dividing by a negative #SwitchDon’t switch-3x < 93x < -12Original Problem(s) x > -3 x < -4Solution Graphing Linear Inequalities: Closed circleOpen Circle320040023495009144002349500 , < , >3048000177800018669002349500125730013779500 Solve the following linear inequalities, then graph each solution:EX 1]3x + 12 < 9EX 2]4x – 3 6x + 15297180013906500-22860013906500EX 3] –4x + 16 > 4EX 4] –3x – 6 3x + 6297180013906500-22860013906500 Graph Compound InequalitiesWhat is different now?2667014795500 EX] -1 < x < 2EX] x -2 or x > 127412952476500 HOW DO I REMEMBER THESE???_________________________ _______________________Graph the following inequalities.1. 2. or -114300107950028575001079500Solve the compound inequality, and then graph your solution.3. 3. or 28575002159000-1143007493000Important things to remember about solving and graphing inequalities:***When to use open circle vs. closed circle***When to switch signs when solving*** hand shake *** *** boat oars ***Solve the following compound inequalities, then graph each solution:EX 1] -10 3x + 5 < 1425527004699000EX 2] 4x + 3 < -13 OR 5x – 5 > -532385001460500Day4: Absolute Value Inequalities We will: solve absolute value inequalitiesI will:KEY: Absolute value turns simple inequalities into compound inequalities because we have to consider the negative case. Less Than: Greater Than: |x| < 3 means |x| > 3 means set of pts whose distance from ____ is _____ set of pts whose distance from ____ is _____ 35890207620000-5048258382000 -3 < x < 3 x < ? 3 or x > 3Write the absolute value inequalities that would correspond with these graphs:-1828805778500448437010922000523494057785004430395704850035871157112000How we DO it: Solve an Absolute Value Inequalities – Isolate the absolute value symbol on one side of the equal signBreak the equation into derived equations – the positive case and the negative case (for the negative case – KEEP, CHANGE, CHANGE)Solve both equationsCheck your solutions (WARNING: There may be extraneous solutions!)Solve and Graph the Absolute Value Inequalities. |x + 3| ≥ 5 Think: ____________ 20891555562500 |3x-2> 18 Think: ____________ 20574057150003. x – 2 > 4 Think: ___________208915-641300 4. 5x + 1 ≥ 16 Think: __________16043296625200How about LESS THAN? 3x + 3 6 Think: ________ 20891546672500 4x – 9 21 Think: ____________ 32956512954000 3. x – 1 < 4 Think: ___________20891567188000 4. 25x + 1 - 4 ≤ 14 Think: ________329565172309200Try these:356552510223500|3x + 1|+ 2 < 8367792099060002|x - 3|- 2 ≥ 836779209906000|x - 3|- 2 < -8THINK about this one! TURN&TALK: Change something about #3 so that…….. ................
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