Loudoun County Public Schools / Overview



3957955-3212100Unit 3: Absolute ValueSchedule of Upcoming ClassesDay 1A 9/21Introducing AV functionsDay 2A9/23Transformations & Analyzing GraphsDay 3A9/27Review of Graphing AV functions *Day 4A9/29Quiz: Graphing & AnalyzingAV FunctionsDay 5A10/3Solving & Graphing AV EquationsDay 6A10/5Solving & Graphing AV InequalitiesDay 7A10/7Unit ReviewDay 8A10/12Unit Test*Skills Review due & Skills CheckSee Ms. Raschiatore AS SOON AS POSSIBLE to get work and any help you need.Blank copies of the notes are on my CMS page. Homework assignments are also on my CMS page if you lose your packet.Need Help?Mu Alpha Theta is available to helpMonday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:15.Ms. Raschiatore will be available to answer questions in the morning beginning at 8:30 (room L404)Need to make up a test/quiz?Math Make Up Room is open Weds mornings and Tues/Thurs afternoons.Schedule is posted on front white board near the door. Algebra 2Name ____________________Unit 3 “Absolute Value Functions”Date ____________ Block ___Day 1: Introducing… The Absolute Value Function ~ Using the calculator to graph and finding key characteristics of the graphright14986000Let’s take a look at y = x (Graph in your calculator as )What happens if we change every negative y-valueto a positive value? i.e. make the point (3, -3) become(3, +3)Does this sound familiar? What takes negative values and makes them positive?Introducing …….. the Absolute Value Function We can analyze the parent function for special points and behavior - 368174215049500 Use your calculator to graph: Domain:Range:Vertex:y-intercept:zeros (roots, x-intercepts, solutions):Increasing:Decreasing:End Behavior:Slope of right branch:We can also move the parent function to other places on the coordinate plane. Use your calculator to graph each and find the key characteristics.9048758572500Domain:Range:Vertex: Y-intercept:Zeros / X-intercepts:Increasing:Decreasing:End Behavior:Slope of right branch:10344158572500Domain:Range:Vertex: Y-intercept:Zeros / X-intercepts:Increasing:Decreasing:End Behavior:Slope of right branch:Are you noticing any patterns yet? Let’s look at domain and range.11636333302000Domain:Range:Vertex: Y-intercept:Zeros / X-intercepts:Increasing:Decreasing:End Behavior:Slope of right branch:12696781968500Domain:Range:Vertex: Y-intercept:Zeros / X-intercepts:Increasing:Decreasing:End Behavior:Slope of right branch:Are you noticing any patterns yet? Let’s look at slope of the right branch.518731512763500Review topic: 1.Graph the inverse of the Absolute Value Function (start out with the original )Think about how you graph an inverse! Is the inverse a function?51746151016000Were you expecting this? Why? 2.Graph an absolute value function that has aremovable discontinuity at (3, 4)Day 2: Graphing Absolute Value FUNctions Using TRANSFORMATIONSIn these notes we will Learn a new technique for graphing a function – shifting it up, down, left, rightSo we can …. Graph absolute value functions WITHOUT a calculator49244251079500Eventually graph ANY function given its parent shape.First, let’s graph the absolute value “parent function”, y = | x |Use your calculator to graph this function in Y1What is the vertex of the graph?Exploration of Transformations – Vertical Shifts456057083820001. Graph y = |x| + 2 on your calculator in Y2.a) Sketch this graph and the “parent function”.b) How does the graph move? (up or down) _____c) What is the vertex of the graph? ______456057036830002. Graph y = |x| - 5 on your calculator in Y2.a) Sketch this graph and the “parent function”.b) How does the graph move? (up or down) _____c) What is the vertex of the graph? ______3. 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?5264150-8794300Exploration 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? ______526923091762002. 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? ______5262245119702003. 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? ______5281683114452004. 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? ______5. 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 ‘taller’ or ‘shorter’?48139353810004. 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? __________4802505173990005. 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?1. __________________________2. __________________________ Examples 1 and 2 Examples 3 and 4Exploration of ALL Transformations –1. Graph y = |x| on your calculator in Y1. 2. Graph y = | x – 3 | – 4 on you calculator in Y2.a) What direction does the graph open? _____b) How does the graph move? (Left/right, up/down) ___________c) What is the vertex of the graph? ______3. Graph y = – | x – 2 | – 3 on you calculator in Y2.a) What direction does the graph open? _____b) How does the graph move? (left/right, up/down) ___________c) What is the vertex of the graph? ______4. Graph y = 2|x + 6| – 4 on you calculator in Y2.a) What direction does the graph open? _____b) How does the graph move? (left/right, up/down) ___________c) What is the vertex of the graph? ______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 Dilation(stretch or shrink of….)1. y = ? |x + 4| - 9 2. y = - 2|x + 1| + 63. y = 4|x – 3| + 54. y = - ? |x – 7| + 35. y = 2|x + 4| - 1 Graph the following using your knowledge of transformations (no calculator). Verify (check your answer) by graphing on your calculator and comparing your answer to the calculators.1. y = ? |x + 4| - 92. y = - 2|x + 1| + 63. y = 4|x – 3| + 5-12890517399021824951644654545330183515Domain: _________Domain: _________ Domain: ________Range: __________Range: __________ Range: ________323469066040x00x110490111125xy00xyGiven the absolute value equation graph, write the absolute value equation: 1.2. y = ___________________ f(x) = __________________205740100965003568065123825003.4. _________________________________________Day 3: Review Graphing Absolute Value Functions(with and without a calculator)1. Which absolute value function(s) open up? (there may be more than one answer!)y = -2|x – 5|B. y = |x + 1| – 7 C. y = -|x + 4| + 8D. y = ?|x – 9|2. Which absolute value function(s) are vertically stretched? (there may be more than one answer!)y = -2|x – 5|B. y = |x + 1| – 7 C. y = -|x + 4| + 8D. y = ?|x – 9|3. Which absolute value function(s) have an absolute minimum at the vertex? (there may be more than one answer!)y = -2|x – 5|B. y = |x + 1| – 7 C. y = -|x + 4| + 8D. y = ?|x – 9|4. Given f(x) = |x + 9|. The vertex of the function moves from (0, 0) nine units ______:A. leftB. rightC. upD. down254635147320xy00xy5. SketchWhat is the range?What is the end behavior?The vertex is… (circle all that apply) A) a relative minimum B) a relative maximum C) an absolute minimum D) an absolute maximum91440278130xy00xy6. Sketch What is the range?What is the end behavior?The vertex is… (circle all that apply) A) a relative minimum B) a relative maximum C) an absolute minimum D) an absolute maximum7. Write the equation for the following absolute value functions….47256703810003790953238500 b)____________________________ ____________________________32194527940004330065571500 c) d) ____________________________ ____________________________e) With a vertex at ( - 8, 2 ) that is vertically stretched by a scale of 2: ____________________________f) That opens downward and has a vertex at (5, 0):____________________________g) That is vertically shrunk and shifted to the right (you pick the numerical details!) ____________________________Day 5: Absolute Value EquationsObjective: To understand the definition of absolute value and to know how to use this definition in solving absolute value equations.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 4”47415452857500Another way – think of two FUNCTIONS. Where are they EQUAL?Graph y=|x| and y=4The solution(s) is/are the x-coordinates of the points of intersection16383001746253. |x - 4| = 3As distance: the set of points whose distance from ____ is equal to ___48501301841500As functions - What two functions are we looking at here?Where are they EQUAL? -241935-17907000How we SOLVE ALGEBRAICALLY: To 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. -3|x - 1| + 2 = – 4 Let’s verify our answers graphically (this is how you can use your calculator to check your hw)Y1 =Y2 =Y1 =Y2 =36175951206500left2540003. |2x + 12| = 4x4. |4x + 5| = 2x + 45. 2 |x + 7| - 5 = 156. -2 |x + 7| - 5 = 15Discuss with a partner: What was different between #5 and 6? What kind of absolute value equation would have no solution? What would the graphs look like if an absolute value equation had no solution?Algebra 2 WARM-UP(before absolute value Inequalities)Review from Algebra 1 ~ solving linear inequalities (We will use this today!) Inequality Symbols: < __________________ , ____________________ > ___________________ , ____________________Don’t forget switch the sign of the inequality when multiplying / dividing by a negative #SwitchDon’t switch313563085725-3x < 93x < -12Original Problem(s)3030855101600 x > -3 x < -4Solution(s) 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 + 6297180013906500-22860013906500D) Graphing Compound Inequalities2667014795500 EX] -1 < x < 2EX] x -2 or x > 127412952476500Graph the following inequalities.1. 2. or 32213553937000-1143001079500Solve the compound inequality, and then graph your solution.3. 4. or 29698952195300-361952603500Day 6: Absolute Value InequalitiesAbsolute value turns simple inequalities into compound inequalities because we have to consider the negative case. Less than:| x | < 3 means:set of points whose distance from ____ is_________________742956667500-3 < x < 3Greater than: | x | > 3 means:set of points whose distance from ____ is__________________723906667500x x < – 3 or x > 3Practice: Write the absolute value inequalities that would correspond with these graphs:-1828805778500448437010922000523494057785004430395704850035871157112000 -2 2 -2 2AND or ORAND or ORDiscuss: What was a key difference between solving linear equations and absolute value equations?How we Analytically Solve an Absolute Value Inequality – Isolate the absolute value symbol on one side of the equal signBreak the equation into 2 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!)Write your answer using interval notationSolve and Graph the Absolute Value Inequality: | x + 3 | ≥ 5AND or OR ? Verify graphically4451985387350011677655969000Graph your solution:3. | x – 2 | > 4 AND or OR ?20891515589700Verify graphically4857759144000Does this look like AND or OR? 4. | 5x + 1 | ≥ 16 AND or OR ?692151651000Verify graphically5759453048000Does this look like AND or OR?. | 5x + 1 | – 4 ≤ 14 AND or OR ?Verify graphically47644051308100037211073723500Try these:356552510223500|3x + 1|+ 2 < 8367792099060002|x - 3|- 2 ≥ 836779209906000|x - 3|- 2 < -8THINK about this one! Discuss with a partner: Change something about #3 so that……It has a solution of all real numbersIt has a solution that is 2 separate inequalities ................
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