Mathematical Practices

Download for free
Find the solutions in the app
Android iOS
Exercises marked with requires Mathleaks premium to view it's solution in the app. Download Mathleaks app on Google Play or iTunes AppStore.
Sections
Monitoring Progress
Exercise name Free?
Monitoring Progress 1 It takes three distinct points to graph a quadratic function. Let's look at a table of values for y=-x2, then plot the points and the parabola.x-x2y=-x2 -2-(-2)2-4 -1-(-1)2-1 0-(0)20 1-(1)2-1 2-(2)2-4 Now, let's plot those points and draw our parabola.This parabola has a highest point at (0,0) and the graph opens downwards.
Monitoring Progress 2 It takes three distinct points to graph a quadratic function. Let's look at a table of values for y=2x2, then plot the points on a graph.x2x2y=2x2 -22(-2)28 -12(-1)22 02(0)20 12(1)22 22(2)28 Now, let's plot those points and draw our curve.This graph has it's lowest point at (0,0) and opens upwards.
Monitoring Progress 3 It takes three distinct points to graph a quadratic function. Let's look at a table of values for f(x)=2x2+1, plot the points, then draw the graph.x2x2+1f(x)=2x2+1 -22(-2)2+19 -12(-1)2+13 02(0)2+11 12(1)2+13 22(2)2+19 Now, let's plot those points and draw our curve.This graph has a lowest point is at (0,1) and opens upwards.
Monitoring Progress 4 It takes three distinct points to graph a quadratic function. Let's look at a table of values for f(x)=2x2−1, plot the points, then look at the graph.x2x2−1f(x)=2x2−1 -22(-2)2−17 -12(-1)2−11 02(0)2−1-1 12(1)2−11 22(2)2−17 Now, let's plot those points and draw our curve.This graph has its lowest point at (0,-1) and opens upwards.
Monitoring Progress 5 It takes three distinct points to graph a quadratic function. Let's look at a table of values for f(x)=21​x2+4x+3, plot the points, and look at the graph.x21​x2+4x+3f(x)=21​x2+4x+3 -821​(-8)2+4(-8)+33 -621​(-6)2+4(-6)+3-3 -421​(-4)2+4(-4)+3-5 -221​(-2)2+4(-2)+3-3 021​(0)2+4(0)+33 Now, let's plot those points and draw our curve.This graph has its lowest point at (-4,-5) and opens upwards.
Monitoring Progress 6 It takes three distinct points to graph a quadratic function. Let's look at a table of values for f(x)=21​x2−4x+3, plot the points, then look at the graph.x21​x2−4x+3f(x)=21​x2−4x+3 021​(0)2−4(0)+33 221​(0)2−4(0)+3-3 421​(4)2−4(4)+3-5 621​(6)2−4(6)+3-3 821​(8)2−4(8)+33 Now, let's plot those points and draw our curve.This quadratic has its lowest point at (4,-5) and opens upwards.
Monitoring Progress 7 It takes three distinct points to graph a quadratic function. Let's look at a table of values for f(x)=-2(x+1)2+1, plot the points, and look at the graph.x-2(x+1)2+1y=-2(x+1)2+1 -3-2(-3+1)2+1-7 -2-2(-2+1)2+1-1 -1-2(-1+1)2+11 0-2(0+1)2+1-1 1-2(1+1)2+1-7 Now, let's plot those points and draw our curve.This quadratic has its highest point at (-1,1) and opens downwards.
Monitoring Progress 8 It takes three distinct points to graph a quadratic function. Let's look at a table of values for f(x)=-2(x+1)2+1, plot the points, and then look at the graph.x-2(x−1)2+1y=-2(x−1)2+1 -1-2(-7−1)2+1-7 0-2(0−1)2+1-1 1-2(1−1)2+11 2-2(2−1)2+1-1 3-2(3−1)2+1-7 Now, let's plot those points and draw our curve.This quadratic its highest point at (1,1) and opens downwards.
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.