函数图像的变换(2)
2013-08-06 03:26
本范例探究函数表达式的变动对函数图像的影响，演示平移、反射等几何变换如何改变函数图形。通过动态交互，用户可观察参数变化时函数 f(x) 与 g(x) 的图像对应关系，理解函数变换的代数与几何意义。支持多语言切换与重置操作，适合中学及大学函数教学使用。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.350.0" id="3317318d-39c5-4017-8b83-870eaeb3290c" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1000" height="690" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.26666666666666666" orientation="0" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 59 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="2" size="496" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="1" size="244" window="100,100,250,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="0" size="184" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="true" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="true" stylebar="true" location="1" size="387" window="87,134,1105,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="7" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="998" height="496"/>
<coordSystem xZero="478.99999999999994" yZero="243.5" scale="39.511261184345315" yscale="39.51126118434534"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="153" b="204"/>
<gridColor r="160" g="160" b="160"/>
<lineStyle axes="3" grid="0"/>
<axis id="0" show="true" label="x" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="y" unitLabel="" tickStyle="1" showNumbers="true"/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="998" height="184"/>
<coordSystem xZero="1079.6568269148981" yZero="209.7028197223785" scale="32.41127720775451" yscale="32.411277207754566"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="250" b="205"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="false" label="x" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="y" unitLabel="" tickStyle="1" showNumbers="true"/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label ="listNum" exp="{3, 4, 5}"/>
<element type="list" label="listNum">
<show object="true" label="false" ev="11"/>
<condition showObject="showPolyPoints"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelOffset x="897" y="113"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectedIndex val="1"/>
<comboBox val="true"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
<caption val=":"/>
</element>
<command name="SelectedElement">
<input a0="listNum"/>
<output a0="n"/>
</command>
<element type="numeric" label="n">
<value val="4"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<element type="point" label="A">
<show object="true" label="true"/>
<condition showObject="showPolyPoints"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-4" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<element type="point" label="B">
<show object="true" label="true"/>
<condition showObject="showPolyPoints"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-2" y="1" z="1"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<element type="point" label="C">
<show object="true" label="true"/>
<condition showObject="showPolyPoints"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="1" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<element type="point" label="D">
<show object="true" label="true"/>
<condition showObject="showPolyPoints ∧ (n ≥ 4)"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="4" y="21"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="2" y="2" z="1"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<element type="point" label="E">
<show object="true" label="true"/>
<condition showObject="showPolyPoints ∧ (n ≥ 5)"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="4" y="1" z="1"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<command name="First">
<input a0="{A, B, C, D, E}" a1="n"/>
<output a0="listPoint"/>
</command>
<element type="list" label="listPoint">
<show object="false" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<expression label ="listTextB" exp="{"$+5$", "$+4$", "$+3$", "$+2$", "$+1$", "$0$", "$-1$", "$-2$", "$-3$", "$-4$", "$-5$"}"/>
<element type="list" label="listTextB">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelOffset x="509" y="15"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectedIndex val="5"/>
<comboBox val="true"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<caption val="-"/>
</element>
<expression label ="listB" exp="{5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5}"/>
<element type="list" label="listB">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<labelOffset x="337" y="20"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
<caption val="|"/>
</element>
<expression label ="listTextA" exp="{"$+4$", "$+3$", "$+2$", "$+1$", "$+\frac{1}{2}$", "$+\frac{1}{3}$", "$+\frac{1}{4}$", "$-\frac{1}{4}$", "$-\frac{1}{3}$", "$-\frac{1}{2}$", "$-1$", "$-2$", "$-3$", "$-4$"}"/>
<element type="list" label="listTextA">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelOffset x="324" y="15"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectedIndex val="3"/>
<comboBox val="true"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<caption val="-"/>
</element>
<expression label ="listA" exp="{4, 3, 2, 1, 0.5, 0.3333333333333333, 0.25, -0.25, -0.3333333333333333, -0.5, -1, -2, -3, -4}"/>
<element type="list" label="listA">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Element" type="numeric">
<input a0="listB" a1="SelectedIndex[listTextB]"/>
<output a0="b"/>
</command>
<element type="numeric" label="b">
<value val="0"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<command name="Element" type="numeric">
<input a0="listA" a1="SelectedIndex[listTextA]"/>
<output a0="a"/>
</command>
<element type="numeric" label="a">
<value val="1"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<expression label ="listTextP" exp="{"$+4$", "$+3$", "$+2$", "$+1$", "$+\frac{1}{2}$", "$+\frac{1}{3}$", "$+\frac{1}{4}$", "$-\frac{1}{4}$", "$-\frac{1}{3}$", "$-\frac{1}{2}$", "$-1$", "$-2$", "$-3$", "$-4$"}"/>
<element type="list" label="listTextP">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelOffset x="172" y="15"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectedIndex val="3"/>
<comboBox val="true"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<caption val="-"/>
</element>
<expression label ="listTextQ" exp="{"$+5$", "$+4$", "$+3$", "$+2$", "$+1$", "$0$", "$-1$", "$-2$", "$-3$", "$-4$", "$-5$"}"/>
<element type="list" label="listTextQ">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelOffset x="626" y="15"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectedIndex val="5"/>
<comboBox val="true"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<caption val="-"/>
</element>
<expression label="textInput" exp=""y=g(x)=\text{ }f(\text{ }x\;+\text{ }) +""/>
<element type="text" label="textInput">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="0"/>
<absoluteScreenLocation x="24" y="9"/>
</element>
<expression label ="listP" exp="{4, 3, 2, 1, 0.5, 0.3333333333333333, 0.25, -0.25, -0.3333333333333333, -0.5, -1, -2, -3, -4}"/>
<element type="list" label="listP">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Element" type="numeric">
<input a0="listP" a1="SelectedIndex[listTextP]"/>
<output a0="p"/>
</command>
<element type="numeric" label="p">
<value val="1"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<expression label ="listQ" exp="{5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5}"/>
<element type="list" label="listQ">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Element" type="numeric">
<input a0="listQ" a1="SelectedIndex[listTextQ]"/>
<output a0="q"/>
</command>
<element type="numeric" label="q">
<value val="0"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<command name="FitPoly">
<input a0="listPoint" a1="n - 1"/>
<output a0="f"/>
</command>
<element type="function" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="9" y="-41"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<caption val="$y=f(x)$"/>
</element>
<expression label="g" exp="g(x) = (p * f((a * x) + b)) + q" />
<element type="function" label="g">
<show object="true" label="false" ev="4"/>
<condition showObject="showGraph ∧ (¬equal)"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-59" y="-42"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<caption val="y=g(x)"/>
</element>
<expression label="textGx" exp=""y=g(x)=" + (LaTeX[If[p ≟ 1, "", If[p ≟ -1, "-", FractionText[p]]]]) + "f \left( " + (LaTeX[If[a ≟ 1, "", If[a ≟ -1, "-", FractionText[a]]]]) + "x" + (LaTeX[If[b > 0, "+", ""]]) + "" + (LaTeX[If[b ≠ 0, Text[b], ""]]) + " \right) " + (LaTeX[If[q > 0, "+", ""]]) + "" + (LaTeX[If[q ≠ 0, Text[q], ""]]) + """ />
<element type="text" label="textGx">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="0"/>
<absoluteScreenLocation x="25" y="86"/>
</element>
<element type="boolean" label="showGraph">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="621" y="140"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="Show graph of y=g(x)"/>
</element>
<element type="boolean" label="showPolyPoints">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="621" y="107"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="Show points defining f(x)"/>
</element>
<element type="boolean" label="showFreePoints">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="621" y="75"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[P,(0,f(0))] SetValue[P',(0,g(0))]"/>
<caption val="Show movable points on graphs"/>
</element>
<expression label="equal" exp="a ≟ 1 ∧ b ≟ 0 ∧ p ≟ 1 ∧ q ≟ 0" />
<element type="boolean" label="equal">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="textGxCopy" exp=""" + (LaTeX[textGx]) + """ />
<element type="text" label="textGxCopy">
<show object="true" label="false"/>
<condition showObject="showGraph"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<absoluteScreenLocation x="12" y="111"/>
</element>
<element type="button" label="buttonReset">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="756" y="15"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[listTextA,4] SetValue[listTextP,4] SetValue[listTextB,6] SetValue[listTextQ,6] SetValue[showGraph,false] SetValue[showFreePoints,false] SetValue[showPolyPoints,false]"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<caption val="Reset"/>
</element>
<element type="button" label="button0">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="3"/>
<labelOffset x="885" y="15"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[lang,If[lang==1,2,1]] RunClickScript[buttonLang] "/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<caption val="中文"/>
</element>
<element type="button" label="buttonLang">
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="3"/>
<labelOffset x="918" y="112"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<ggbscript val="SetCaption[button0,If[lang==1,"ENG","中文"]] SetCaption[showFreePoints,If[lang==1,"顯示圖像上的動點","Show movable points on graphs"]] SetCaption[showPolyPoints,If[lang==1,"顯示定義f(x)的點","Show points defining f(x)"]] SetCaption[showGraph,If[lang==1,"顯示 y=g(x) 的圖像","Show graph of y=g(x)"]] SetCaption[buttonReset,If[lang==1,"重設","Reset"]] "/>
<font serif="false" sizeM="2" size="20" style="0"/>
</element>
<element type="numeric" label="lang">
<value val="2"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="3"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Join">
<input a0="{{Extremum[f]}, {Root[f]}, {(0, f(0))}}"/>
<output a0="list1"/>
</command>
<element type="list" label="list1">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="3"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Join">
<input a0="{{Extremum[g]}, {Root[g]}, {(0, g(0))}}"/>
<output a0="list2"/>
</command>
<element type="list" label="list2">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="3"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="P_0"/>
</command>
<element type="point" label="P_0">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="3"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="0.03412087473291703" y="-0.34132497913844473" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="ClosestPoint">
<input a0="list1" a1="P_0"/>
<output a0="P_1"/>
</command>
<element type="point" label="P_1">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="3"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="0" y="-0.3333333333333333" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="g"/>
<output a0="Q_0"/>
</command>
<element type="point" label="Q_0">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="3"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="-7.999999999999897" y="-32.999999999998316" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="ClosestPoint">
<input a0="list2" a1="Q_0"/>
<output a0="Q_1"/>
</command>
<element type="point" label="Q_1">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="3"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="-4" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="d" exp="AxisStepX[] / 10" />
<element type="numeric" label="d">
<value val="0.1"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="DynamicCoordinates">
<input a0="P_0" a1="If[Distance[P_0, P_1] < d, x(P_1), x(P_0)]" a2="If[Distance[P_0, P_1] < d, y(P_1), y(P_0)]"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true"/>
<condition showObject="showFreePoints"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="15" y="39"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="0" y="-0.3333333333333333" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="DynamicCoordinates">
<input a0="Q_0" a1="If[Distance[Q_0, Q_1] < d, x(Q_1), x(Q_0)]" a2="If[Distance[Q_0, Q_1] < d, y(Q_1), y(Q_0)]"/>
<output a0="P'"/>
</command>
<element type="point" label="P'">
<show object="true" label="true"/>
<condition showObject="showGraph ∧ showFreePoints ∧ (¬equal)"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelOffset x="-1" y="34"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="-7.999999999999897" y="-32.999999999998316" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
</construction>
</geogebra>