Exploring Absolute Value Functions and Inequalities
2020-03-09 23:58
This GeoGebra applet dynamically demonstrates absolute value functions and related inequalities. By adjusting parameters a, h, and k, users observe function transformations. It visually illustrates solution sets for strict and non-strict inequalities. Using Locus and If commands, it highlights regions satisfying the conditions, helping students grasp algebraic-geometric synthesis.
In collections
Secondary Vocational Mathematics
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="w" id="95dd08ff-aafc-49f8-a767-b2cb5eb48cdc" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1339" height="566" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.2083644510828977" orientation="1" />
</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 51 | 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 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="362" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1339" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="279" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" 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="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" 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="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1339" height="566"/>
<coordSystem xZero="280.11019910348637" yZero="273.12044572061797" scale="37.10214632681543" yscale="37.10214632681535"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="dUL">
<show object="true" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<auxiliary val="true"/>
<coords x="-7.390782353977023" y="7.197284239281268" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="dDR">
<show object="true" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<auxiliary val="true"/>
<coords x="7.204282894129527" y="-7.345468947004106" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="幕布" exp="幕布(x, y) = x ≤ x(dUL) ∨ x ≥ x(dDR) ∨ y ≤ y(dDR) ∨ y ≥ y(dUL)" />
<element type="functionNVar" label="幕布">
<show object="true" label="false" ev="4"/>
<objColor r="224" g="224" b="224" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Vector">
<input a0="(x(dUL), 0)" a1="(x(dDR), 0)"/>
<output a0="轴x"/>
</command>
<element type="vector" label="轴x">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="14.59506524810655" y="0" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint x="-7.390782353977023" y="0" z="1"/>
</element>
<command name="Vector">
<input a0="(0, y(dDR))" a1="(0, y(dUL))"/>
<output a0="轴y"/>
</command>
<element type="vector" label="轴y">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0" y="14.542753186285374" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint x="0" y="-7.345468947004106" z="1"/>
</element>
<element type="numeric" label="a">
<value val="3"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="80" x="28" y="59" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="h">
<value val="2"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="80" x="30" y="111" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="k">
<value val="-3"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="80" x="30" y="164" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="f" exp="f(x) = (a * abs(x - h)) + k" />
<element type="function" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="85" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="LaTeX">
<input a0="f" a1="true" a2="true"/>
<output a0="text3"/>
</command>
<element type="text" label="text3">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="565" y="101"/>
</element>
<command name="Solutions">
<input a0="f(x) = 0"/>
<output a0="jj解集"/>
</command>
<element type="list" label="jj解集">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
<symbolic val="true" />
</element>
<command name="Point">
<input a0="Segment[(x(dUL), 0), (x(dDR), 0)]"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<auxiliary val="true"/>
<coords x="-1.1571334811205283" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="If">
<input a0="f(x(A)) < y(dUL) ∧ f(x(A)) > 0" a1="f(x(A))"/>
<output a0="yUp"/>
</command>
<element type="numeric" label="yUp">
<value val="6.471400443361585"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<auxiliary val="true"/>
</element>
<command name="Locus">
<input a0="(x(A), yUp)" a1="A"/>
<output a0="loc1"/>
</command>
<element type="locus" label="loc1">
<show object="true" label="false" ev="4"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="If">
<input a0="f(x(A)) > y(dDR) ∧ f(x(A)) < 0" a1="f(x(A))"/>
<output a0="yDown"/>
</command>
<element type="numeric" label="yDown">
<value val="NaN"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<auxiliary val="true"/>
</element>
<command name="Locus">
<input a0="(x(A), yDown)" a1="A"/>
<output a0="loc2"/>
</command>
<element type="locus" label="loc2">
<show object="true" label="false" ev="4"/>
<objColor r="220" g="20" b="60" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="dcz" exp="x = h" type="line" />
<element type="line" label="dcz">
<show object="true" label="true" ev="4"/>
<objColor r="160" g="160" b="160" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<coords x="1" y="0" z="-2"/>
<lineStyle thickness="5" type="30" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label="x1" exp="(jj解集(1), 0)" type="point" />
<element type="point" label="x1">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-30" y="7"/>
<labelMode val="3"/>
<coords x="1" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large x_1$"/>
</element>
<expression label="x2" exp="(jj解集(2), 0)" type="point" />
<element type="point" label="x2">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-1" y="5"/>
<labelMode val="3"/>
<coords x="3" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large x_2$"/>
</element>
<expression label="txtfD0" exp=""" + (LaTeX[f]) + ">0"" />
<element type="text" label="txtfD0">
<symbolic val="true" />
<show object="false" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<auxiliary val="true"/>
<startPoint x="8.620496283842076" y="0.38067609732498386" z="1"/>
</element>
<expression label="txtfX0" exp=""" + (LaTeX[f]) + "<0"" />
<element type="text" label="txtfX0">
<symbolic val="true" />
<show object="false" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<auxiliary val="true"/>
<startPoint x="8.365215169124415" y="0.23885325581517108" z="1"/>
</element>
<expression label="xlfD0" exp="{txtfD0, If[a > 0, If[k ≤ 0, (((("(-∞," * jj解集(1)) * ")∪(") * jj解集(2)) * ",+∞)"), "R"], If[k > 0, (((("(" * jj解集(1)) * ",") * jj解集(2)) * ")"), "∅"]]}" />
<element type="list" label="xlfD0">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
<symbolic val="true" />
</element>
<expression label="xlfX0" exp="{txtfX0, If[a > 0, If[k < 0, (((("(" * jj解集(1)) * ",") * jj解集(2)) * ")"), "∅"], If[k ≥ 0, (((("(-∞," * jj解集(1)) * ")∪(") * jj解集(2)) * ",+∞)"), "R"]]}" />
<element type="list" label="xlfX0">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
<symbolic val="true" />
</element>
<expression label="xl标题" exp="{"不等式", "解集"}"/>
<element type="list" label="xl标题">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
<symbolic val="true" />
</element>
<command name="TableText">
<input a0="xl标题" a1="xlfD0" a2="xlfX0" a3=""|_c""/>
<output a0="txt不含等号"/>
</command>
<element type="text" label="txt不含等号">
<show object="true" label="true" ev="8"/>
<condition showObject="a ≠ 0"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="583" y="264"/>
</element>
<expression label="txtfDD0" exp=""" + (LaTeX[f]) + "≥0"" />
<element type="text" label="txtfDD0">
<symbolic val="true" />
<show object="false" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<auxiliary val="true"/>
<startPoint x="8.195027759312639" y="-2.7961555524948225" z="1"/>
</element>
<expression label="txtfXD0" exp=""" + (LaTeX[f]) + "≤0"" />
<element type="text" label="txtfXD0">
<symbolic val="true" />
<show object="false" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<auxiliary val="true"/>
<startPoint x="8.365215169124415" y="-3.9307382845733247" z="1"/>
</element>
<expression label="xlfDD0" exp="{txtfDD0, If[a > 0, If[k < 0, (((("(-∞," * jj解集(1)) * "]∪[") * jj解集(2)) * ",+∞)"), "R"], If[k > 0, (((("[" * jj解集(1)) * ",") * jj解集(2)) * "]"), If[k ≟ 0, (("\{" * jj解集(1)) * "\}"), "∅"]]]}" />
<element type="list" label="xlfDD0">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
<symbolic val="true" />
</element>
<expression label="xlfXD0" exp="{txtfXD0, If[a > 0, If[k < 0, (((("[" * jj解集(1)) * ",") * jj解集(2)) * "]"), If[k ≟ 0, (("\{" * jj解集(1)) * "\}"), "∅"]], If[k > 0, (((("(-∞," * jj解集(1)) * "]∪[") * jj解集(2)) * ",+∞)"), "R"]]}" />
<element type="list" label="xlfXD0">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
<symbolic val="true" />
</element>
<command name="TableText">
<input a0="xl标题" a1="xlfDD0" a2="xlfXD0" a3=""|_c""/>
<output a0="txt含等号"/>
</command>
<element type="text" label="txt含等号">
<show object="true" label="true" ev="4"/>
<condition showObject="a ≠ 0"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="581" y="391"/>
</element>
<expression label="text5" exp=""对应方程：" + (LaTeX[f]) + "=0\\
的解集为：" + (LaTeX[jj解集]) + """ />
<element type="text" label="text5">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<auxiliary val="true"/>
<absoluteScreenLocation x="592" y="135"/>
</element>
<expression label="text4" exp=""含绝对值的不等式与函数""/>
<element type="text" label="text4">
<show object="true" label="true" ev="40"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<bgColor r="255" g="0" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<absoluteScreenLocation x="574" y="23"/>
</element>
<command name="Text">
<input a0=""y"" a1="(0, y(dUL) - 0.5)" a2="true" a3="true"/>
<output a0="text2"/>
</command>
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<auxiliary val="true"/>
<startPoint x="0" y="6.697284239281268" z="1"/>
</element>
<command name="Text">
<input a0=""x"" a1="(x(dDR) - 0.5, 0)" a2="true" a3="true"/>
<output a0="text1"/>
</command>
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<auxiliary val="true"/>
<startPoint x="6.704282894129527" y="0" z="1"/>
</element>
</construction>
</geogebra>