Construction and Demonstration of a Sawtooth Function
2014-04-08 06:11
This applet demonstrates the input and processing of a sawtooth function, along with the basic usage of the Sequence[] command in GeoGebra. By defining the left and right limits and piecewise expressions of the function, the sequence command is used to generate a sawtooth wave graph, helping to understand the application of piecewise functions and sequence commands in graphing.
In collections
函數
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.4" version="4.4.22.0" id="229e44ca-75ad-41c7-9d49-75d1cd705784" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.28625" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="740,350,700,550" />
<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="false" 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 || 66 68 || 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="true" location="1" size="789" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="229" window="100,100,250,400" />
<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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 | 1 501 67 , 5 19 , 72 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="789" height="463"/>
<coordSystem xZero="391.65073094033374" yZero="269.1890828190411" scale="65.44142760791317" yscale="65.44145432170967"/>
<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="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="0" grid="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0"/>
<grid distX="1.0" distY="1.0" distTheta="0.5235987755982988"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
<startAnimation val="true"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label ="f" exp="f(x) = floor(x)"/>
<element type="function" label="f">
<show object="true" label="false"/>
<condition showObject="t ≟ 0"/>
<objColor r="220" g="20" b="60" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<expression label="xmin" exp="floor(x(Corner[1]))" />
<element type="numeric" label="xmin">
<value val="-7.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="xmax" exp="floor(x(Corner[2]))" />
<element type="numeric" label="xmax">
<value val="6.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
<input a0="(k, k)" a1="k" a2="xmin" a3="xmax"/>
<output a0="fL"/>
</command>
<element type="list" label="fL">
<show object="true" label="false"/>
<condition showObject="t ≟ 0"/>
<objColor r="220" g="20" b="60" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="(k + 1, k)" a1="k" a2="xmin" a3="xmax"/>
<output a0="fR"/>
</command>
<element type="list" label="fR">
<show object="true" label="false"/>
<condition showObject="t ≟ 0"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label ="g" exp="g(x) = x"/>
<element type="function" label="g">
<show object="true" label="false"/>
<condition showObject="t ≟ 0"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
</element>
<expression label="f_1" exp="f_1(x) = g(x) - f(x)" />
<element type="function" label="f_1">
<show object="true" label="false"/>
<condition showObject="t ≟ 1"/>
<objColor r="51" g="153" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
</element>
<element type="numeric" label="t">
<value val="4.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="1"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="100.0" x="76.0" y="52.0" fixed="true" horizontal="true"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="true"/>
</element>
<expression label="f_2" exp="f_2(x) = f_1(x) - 1 / 2" />
<element type="function" label="f_2">
<show object="true" label="false"/>
<condition showObject="t ≟ 2"/>
<objColor r="0" g="153" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
</element>
<expression label="f_3" exp="f_3(x) = abs(f_2(x))" />
<element type="function" label="f_3">
<show object="true" label="false"/>
<condition showObject="t ≟ 3"/>
<objColor r="0" g="51" b="204" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
</element>
<command name="Sequence">
<input a0="(k, 0)" a1="k" a2="xmin" a3="xmax"/>
<output a0="f_1L"/>
</command>
<element type="list" label="f_1L">
<show object="true" label="false"/>
<condition showObject="t ≟ 1"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="(k + 1, 1)" a1="k" a2="xmin" a3="xmax"/>
<output a0="f_1R"/>
</command>
<element type="list" label="f_1R">
<show object="true" label="false"/>
<condition showObject="t ≟ 1"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="(k, (-1) / 2)" a1="k" a2="xmin" a3="xmax"/>
<output a0="f_2L"/>
</command>
<element type="list" label="f_2L">
<show object="true" label="false"/>
<condition showObject="t ≟ 2"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="(k + 1, 1 / 2)" a1="k" a2="xmin" a3="xmax"/>
<output a0="f_2R"/>
</command>
<element type="list" label="f_2R">
<show object="true" label="false"/>
<condition showObject="t ≟ 2"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="(k + 1, 1 / 2)" a1="k" a2="xmin" a3="xmax"/>
<output a0="f_3R"/>
</command>
<element type="list" label="f_3R">
<show object="true" label="false"/>
<condition showObject="t ≟ 3"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="(k + 1, 1 / 2)" a1="k" a2="xmin" a3="xmax"/>
<output a0="f_3L"/>
</command>
<element type="list" label="f_3L">
<show object="true" label="false"/>
<condition showObject="t ≟ 3"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<expression label="f_4" exp="f_4(x) = (2 * f_3(x / 2))" />
<element type="function" label="f_4">
<show object="true" label="false"/>
<condition showObject="t ≥ 4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<command name="LaTeX">
<input a0="f" a1="true" a2="true"/>
<output a0="text1"/>
</command>
<element type="text" label="text1">
<show object="true" label="false"/>
<condition showObject="t ≟ 0"/>
<objColor r="204" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="73" y="72"/>
</element>
<command name="LaTeX">
<input a0="g" a1="true" a2="true"/>
<output a0="text2"/>
</command>
<element type="text" label="text2">
<show object="true" label="false"/>
<condition showObject="t ≟ 0"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="73" y="138"/>
</element>
<command name="LaTeX">
<input a0="f_1" a1="true" a2="true"/>
<output a0="text3"/>
</command>
<element type="text" label="text3">
<show object="true" label="false"/>
<condition showObject="t ≟ 1"/>
<objColor r="51" g="153" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="73" y="72"/>
</element>
<command name="LaTeX">
<input a0="f_2" a1="true" a2="true"/>
<output a0="text4"/>
</command>
<element type="text" label="text4">
<show object="true" label="false"/>
<condition showObject="t ≟ 2"/>
<objColor r="0" g="153" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="73" y="72"/>
</element>
<command name="LaTeX">
<input a0="f_3" a1="true" a2="true"/>
<output a0="text5"/>
</command>
<element type="text" label="text5">
<show object="true" label="false"/>
<condition showObject="t ≟ 3"/>
<objColor r="0" g="51" b="204" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="73" y="72"/>
</element>
<command name="LaTeX">
<input a0="f_4" a1="true" a2="true"/>
<output a0="text6"/>
</command>
<element type="text" label="text6">
<show object="true" label="false"/>
<condition showObject="t ≥ 4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="73" y="72"/>
</element>
</construction>
</geogebra>