#### Mathematical Practices

Find the solutions in the app
Monitoring Progress 9 When an exercise asks a broad question to compare many items like this one, focus on just a couple of characteristics to compare them. Let's consider the direction of the parabolas in the eight exercises and the position of the highest and lowest points.window.JXQ = window.JXQ || []; window.JXQtable = window.JXQtable || {}; if(!window.JXQtable["solution_262830522_1_1586897767_l"]) { window.JXQ.push( function () { var code = function (gid) { var images = []; try { var b = mlg.board([-9.5, 8.5, 9.5, -8.5],{desktopSize:'medium','style':'usa'}); var el = {}; b.xaxis(2,1); b.yaxis(2,1);/*default display*/ el.func1 = b.func('-x^2',{strokeColor:'red'}); el.func2 = b.func('2*x^2',{strokeColor:'green'}); el.func3 = b.func('2*x^2+1',{strokeColor:'teal'}); el.func4 = b.func('2*x^2-1',{strokeColor:'lime'}); el.func5 = b.func('0.5*x^2+4*x+3',{strokeColor:'olive'}); el.func6 = b.func('0.5*x^2-4*x+3',{strokeColor:'darkgray'}); el.func7 = b.func('-2*(x+1)^2+1',{strokeColor:'darkorange'}); el.func8 = b.func('-2*(x-1)^2+1',{strokeColor:'magenta'});/*A*/ /*downward facing*/ mlg.af("showA",function() { b.remove(el); el.func1 = b.func('-x^2',{strokeColor:'red'}); el.legend1 = b.legend(el.func1,[-5,7],"\\textcolor{grey}{1.} y=\\text{-} x^2"); el.func7 = b.func('-2*(x+1)^2+1',{strokeColor:'darkorange'}); el.legend7 = b.legend(el.func7,[-5,5],"\\textcolor{grey}{7.}f(x)=\\text{-}2(x+1)^2+1"); el.func8 = b.func('-2*(x-1)^2+1',{strokeColor:'magenta'}); el.legend8 = b.legend(el.func8,[-5,3],"\\textcolor{grey}{8.}f(x)=\\text{-}2(x-1)^2+1"); });/*B*/ /*upward facing*/ mlg.af("showB",function() { b.remove(el); el.func2 = b.func('2*x^2',{strokeColor:'green'}); el.legend2 = b.legend(el.func2,[-5,5],"\\textcolor{grey}{2.}y=2x^2");el.func3 = b.func('2*x^2+1',{strokeColor:'teal'}); el.legend3 = b.legend(el.func3,[5,5],"\\textcolor{grey}{3.}f(x)=2x^2+1");el.func4 = b.func('2*x^2-1',{strokeColor:'lime'}); el.legend4 = b.legend(el.func4,[5,3],"\\textcolor{grey}{4.}f(x)=2x^2-1");el.func5 = b.func('0.5*x^2+4*x+3',{strokeColor:'olive'}); el.legend5 = b.legend(el.func5,[-5,-6],"\\textcolor{grey}{5.}f(x)=\\frac{1}{2}x^2+4x+3");el.func6 = b.func('0.5*x^2-4*x+3',{strokeColor:'darkgray'}); el.legend = b.legend(el.func6,[5,-6],"\\textcolor{grey}{6.}f(x)=\\frac{1}{2}x^2-4x+3"); });} catch( e ) { mw.hook("ml.error.report").fire(e, { tags: { module: "jsxg", graph: "solution_262830522_1_1586897767_l" } }); console.error(e); } if(b) {b.board.update()} }; mw.loader.using('ext.MLJSXGraph', function() { window.ml.jsxgraph.buildRealGraph("solution_262830522_1_1586897767_l", "solution_262830522_1_1586897767_p", 1, code); }); } ); } window.JXQtable["solution_262830522_1_1586897767_l"] = true;Downwardwindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1242919379_1498458937').on('touchstart mousedown', function () { try { mlg.cf("showA"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );Upwardwindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1910075609_347805097').on('touchstart mousedown', function () { try { mlg.cf("showB"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );We can see that we have three downward facing parabolas, 1, 7, and 8. They all have a negative quadratic term. We can also see that exercises 2, 3, 4, 5, and 6 are facing upwards and have a positive quadratic term.window.JXQ = window.JXQ || []; window.JXQtable = window.JXQtable || {}; if(!window.JXQtable["solution_262830522_2_1475686667_l"]) { window.JXQ.push( function () { var code = function (gid) { var images = []; try { var b = mlg.board([-9.5, 8.5, 9.5, -8.5],{desktopSize:'medium','style':'usa'}); var el = {}; b.xaxis(2,1); b.yaxis(2,1);/*default display*/ el.func1 = b.func('-x^2',{strokeColor:'red'}); el.func2 = b.func('2*x^2',{strokeColor:'green'}); el.func3 = b.func('2*x^2+1',{strokeColor:'teal'}); el.func4 = b.func('2*x^2-1',{strokeColor:'lime'}); el.func5 = b.func('0.5*x^2+4*x+3',{strokeColor:'olive'}); el.func6 = b.func('0.5*x^2-4*x+3',{strokeColor:'darkgray'}); el.func7 = b.func('-2*(x+1)^2+1',{strokeColor:'darkorange'}); el.func8 = b.func('-2*(x-1)^2+1',{strokeColor:'magenta'});/*C*/ /*vertex at origin*/ mlg.af("showC",function() { b.remove(el); el.func1 = b.func('-x^2',{strokeColor:'red'}); el.legend1 = b.legend(el.func1,[-5,-7],"\\textcolor{grey}{1.} y=\\text{-} x^2"); el.func2 = b.func('2*x^2',{strokeColor:'green'}); el.legend2 = b.legend(el.func2,[-5,5],"\\textcolor{grey}{2.}y=2x^2"); }); /*D*/ /*vertex at above*/ mlg.af("showD",function() { b.remove(el); el.func3 = b.func('2*x^2+1',{strokeColor:'teal'}); el.legend3 = b.legend(el.func3,[5,5],"\\textcolor{grey}{3.}f(x)=2x^2+1"); el.func7 = b.func('-2*(x+1)^2+1',{strokeColor:'darkorange'}); el.legend7 = b.legend(el.func7,[-5,-5],"\\textcolor{grey}{7.}f(x)=\\text{-}2(x+1)^2+1"); el.func8 = b.func('-2*(x-1)^2+1',{strokeColor:'magenta'}); el.legend8 = b.legend(el.func8,[5,-5],"\\textcolor{grey}{8.}f(x)=\\text{-}2(x-1)^2+1"); });/*E*/ /*vertex at above*/ mlg.af("showE",function() { b.remove(el); el.func4 = b.func('2*x^2-1',{strokeColor:'lime'}); el.legend4 = b.legend(el.func4,[5,3],"\\textcolor{grey}{4.}f(x)=2x^2-1");el.func5 = b.func('0.5*x^2+4*x+3',{strokeColor:'olive'}); el.legend5 = b.legend(el.func5,[-5,-6],"\\textcolor{grey}{5.}f(x)=\\frac{1}{2}x^2+4x+3");el.func6 = b.func('0.5*x^2-4*x+3',{strokeColor:'darkgray'}); el.legend = b.legend(el.func6,[5,-6],"\\textcolor{grey}{6.}f(x)=\\frac{1}{2}x^2-4x+3"); });} catch( e ) { mw.hook("ml.error.report").fire(e, { tags: { module: "jsxg", graph: "solution_262830522_2_1475686667_l" } }); console.error(e); } if(b) {b.board.update()} }; mw.loader.using('ext.MLJSXGraph', function() { window.ml.jsxgraph.buildRealGraph("solution_262830522_2_1475686667_l", "solution_262830522_2_1475686667_p", 1, code); }); } ); } window.JXQtable["solution_262830522_2_1475686667_l"] = true;high/lowat originwindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1033015978_2085351454').on('touchstart mousedown', function () { try { mlg.cf("showC"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );high/lowabove x-axiswindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn650422666_181662735').on('touchstart mousedown', function () { try { mlg.cf("showD"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );high/lowbelow x-axiswindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1479657226_361149350').on('touchstart mousedown', function () { try { mlg.cf("showE"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );From this graph, we can see that two graphs, 1 and 2, have their highest or lowest point at the origin. Graphs 3, 7, and 8, have their highest or lowest points above the x-axis. And graphs 4, 5, and 6, have their lowest points below the x-axis. Extra info Looking at More Characteristics If we want to look at these graphs with a little more detail, let's consider them a group at at time and compare them to the example function y=x2. Notice that there are similarities in the algebraic representation some of the functions.Group AGroup BGroup CGroup D 1. y=-x22. y=2x23. f(x)=2x2+14. f(x)=2x2−15. f(x)=21​x2+4x+36. f(x)=21​x2−4x+37. f(x)=-2(x+1)2+18. f(x)=-2(x−1)2+1 So let's look at these graphs as groups, then compare the groups.window.JXQ = window.JXQ || []; window.JXQtable = window.JXQtable || {}; if(!window.JXQtable["solution_262830522_extra_1095437825_l"]) { window.JXQ.push( function () { var code = function (gid) { var images = []; try { var b = mlg.board([-9.5, 8.5, 9.5, -8.5],{desktopSize:'medium','style':'usa'}); var el = {}; b.xaxis(2,1); b.yaxis(2,1); /*default display*/ el.func0 = b.func('x^2',{strokeColor:'blue'}); el.legend0 = b.legend(el.func0,[5,7],"\\textcolor{grey}{\\text{example}} y=x^2"); /*A*/ /*Group A*/ mlg.af("showA",function() { b.remove(el); el.func0 = b.func('x^2',{strokeColor:'blue'}); el.legend0 = b.legend(el.func0,[5,7],"\\textcolor{grey}{\\text{example}} y=x^2"); el.func1 = b.func('-x^2',{strokeColor:'red'}); el.legend1 = b.legend(el.func1,[5,-7],"\\textcolor{grey}{1.} y=\\text{-} x^2"); }); /*B*/ /*Group B*/ mlg.af("showB",function() { b.remove(el); el.func0 = b.func('x^2',{strokeColor:'blue'}); el.legend0 = b.legend(el.func0,[-5,5],"\\textcolor{grey}{\\text{example}} y=x^2"); el.func2 = b.func('2*x^2',{strokeColor:'green'}); el.legend2 = b.legend(el.func2,[5,7],"\\textcolor{grey}{2.}y=2x^2"); el.func3 = b.func('2*x^2+1',{strokeColor:'teal'}); el.legend3 = b.legend(el.func3,[5,5],"\\textcolor{grey}{3.}f(x)=2x^2+1"); el.func4 = b.func('2*x^2-1',{strokeColor:'lime'}); el.legend4 = b.legend(el.func4,[5,3],"\\textcolor{grey}{4.}f(x)=2x^2-1"); }); /*C*/ /*Group C*/ mlg.af("showC",function() { b.remove(el); el.func0 = b.func('x^2',{strokeColor:'blue'}); el.legend0 = b.legend(el.func0,[5,7],"\\textcolor{grey}{\\text{example}} y=x^2"); el.func5 = b.func('0.5*x^2+4*x+3',{strokeColor:'olive'}); el.legend5 = b.legend(el.func5,[-5,-6],"\\textcolor{grey}{5.}f(x)=\\frac{1}{2}x^2+4x+3"); el.func6 = b.func('0.5*x^2-4*x+3',{strokeColor:'darkgray'}); el.legend6 = b.legend(el.func6,[5,-6],"\\textcolor{grey}{6.}f(x)=\\frac{1}{2}x^2-4x+3"); }); /*D*/ /*Group D*/ mlg.af("showD",function() { b.remove(el); el.func0 = b.func('x^2',{strokeColor:'blue'}); el.legend0 = b.legend(el.func0,[5,7],"\\textcolor{grey}{\\text{example}} y=x^2"); el.func7 = b.func('-2*(x+1)^2+1',{strokeColor:'darkorange'}); el.legend7 = b.legend(el.func7,[-5,-5],"\\textcolor{grey}{7.}f(x)=\\text{-}2(x+1)^2+1"); el.func8 = b.func('-2*(x-1)^2+1',{strokeColor:'magenta'}); el.legend8 = b.legend(el.func8,[5,-5],"\\textcolor{grey}{8.}f(x)=\\text{-}2(x-1)^2+1"); }); } catch( e ) { mw.hook("ml.error.report").fire(e, { tags: { module: "jsxg", graph: "solution_262830522_extra_1095437825_l" } }); console.error(e); } if(b) {b.board.update()} }; mw.loader.using('ext.MLJSXGraph', function() { window.ml.jsxgraph.buildRealGraph("solution_262830522_extra_1095437825_l", "solution_262830522_extra_1095437825_p", 1, code); }); } ); } window.JXQtable["solution_262830522_extra_1095437825_l"] = true;Group A window.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1797191834_443045347').on('touchstart mousedown', function () { try { mlg.cf("showA"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );Group B window.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1292098135_759977089').on('touchstart mousedown', function () { try { mlg.cf("showB"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );Group C window.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1437164175_638592785').on('touchstart mousedown', function () { try { mlg.cf("showC"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );Group D window.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1374258527_716020758').on('touchstart mousedown', function () { try { mlg.cf("showD"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );Group A1. is in the opposite direction of the example. Group B2,3, and 4 are narrower than the example. 3 is shifted up and 4 is shifted down. Group C5 and 6 are wider than the example. 5's low point is shifted down and to the left. 6's low point is shifted down and to the right. Group D7 and 8 are narrower than the example and facing the opposite direction. 7's high point is shifted up and to the left. 8's high point is shifted up and to the right.