三角函数y=Asin(ωx+φ)的图像与参数变换
2018-12-10 08:46
本范例展示三角函数y=Asin(ωx+φ)的图像，探究参数A（振幅）、ω（角频率）、φ（初相）对图像的影响。通过动态调整参数，直观观察函数图像的平移与伸缩等几何变换，帮助深入理解三角函数的性质与图像特征。
GeoGebra 画板及绘图内容加载中，请稍等……
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.514.0" app="classic" platform="d" id="d3e40a06-e88e-40f7-9d69-6d43a18c600e" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1920" height="1032" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.096875" 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 | 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="557" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1700" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="186" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" 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="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1700" height="850"/>
<coordSystem xZero="494.2402861072516" yZero="427.07027436861074" scale="150.60834111311715" yscale="150.60839776744072"/>
<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" tickDistance="1.0471975511965976" tickExpression="π / 3"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="A">
<value val="1.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.10000000149011612"/>
<bgColor r="0" g="0" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="35.0" y="26.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="ω">
<value val="1.0"/>
<show object="true" label="true"/>
<objColor r="255" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="200.0" x="29.0" y="69.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
<javascript onUpdate="
"/>
</element>
<element type="numeric" label="φ">
<value val="0.0"/>
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="0" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="24.0" y="116.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<expression label="f" exp="f: y = sin(x)" type="function"/>
<element type="function" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.6609396991614873" y="0.46235824432338846" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="C" exp="((x(P) - φ) / ω, y(P))" type="point" />
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<condition showObject="ω ≠ 1"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.6609396991614873" y="0.46235824432338846" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="D" exp="((x(P) - φ) / ω, (A * y(P)))" type="point" />
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<condition showObject="A ≠ 1"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.6609396991614873" y="0.46235824432338846" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="h" exp="h(x) = sin((ω * x) + φ)" />
<element type="function" label="h">
<show object="true" label="true" ev="4"/>
<condition showObject="ω ≠ 1"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="g" exp="g(x) = (A * sin((ω * x) + φ))" />
<element type="function" label="g">
<show object="true" label="true" ev="4"/>
<condition showObject="A ≠ 1"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="Vector">
<input a0="C" a1="D"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="0.0"/>
<lineStyle thickness="6" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="C"/>
</element>
<element type="boolean" label="m">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="301" y="2"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="B" exp="(x(P) - φ, y(P))" type="point" />
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<condition showObject="φ ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.6609396991614873" y="0.46235824432338846" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="B" a1="C"/>
<output a0="w"/>
</command>
<element type="vector" label="w">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="0" b="204" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="0.0"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="216"/>
<coordStyle style="cartesian"/>
<startPoint exp="B"/>
</element>
<expression label="p" exp="p(x) = sin(x + φ)" />
<element type="function" label="p">
<show object="true" label="true" ev="4"/>
<condition showObject="φ ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="text1" exp=""y=Asin(ωx+φ)=" + (LaTeX[A]) + "sin(" + (LaTeX[ω]) + "x+" + (LaTeX[φ]) + ")"" />
<element type="text" label="text1">
<show object="true" label="true" ev="8"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<algebra labelVisible="false"/> <isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<absoluteScreenLocation x="1" y="155"/>
</element>
<command name="Vector">
<input a0="P" a1="B"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<algebra labelVisible="false"/> <coords x="0.0" y="0.0" z="0.0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="P"/>
</element>
<expression label="text3" exp=""B(" + (LaTeX[x(B)]) + "," + (LaTeX[y(B)]) + ")"" />
<element type="text" label="text3">
<show object="true" label="true" ev="8"/>
<condition showObject="φ ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<algebra labelVisible="false"/> <isLaTeX val="true"/>
<font serif="false" sizeM="2.0" size="16" style="0"/>
<absoluteScreenLocation x="697" y="71"/>
</element>
<expression label="text2" exp=""P(" + (LaTeX[x(P)]) + " ," + (LaTeX[y(P)]) + " )"" />
<element type="text" label="text2">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<algebra labelVisible="false"/> <isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="16" style="0"/>
<absoluteScreenLocation x="696" y="30"/>
</element>
<expression label="text4" exp=""C(" + (LaTeX[x(C)]) + "," + (LaTeX[y(C)]) + ")"" />
<element type="text" label="text4">
<show object="true" label="true" ev="8"/>
<condition showObject="ω ≠ 1"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<algebra labelVisible="false"/> <isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="16" style="0"/>
<absoluteScreenLocation x="696" y="110"/>
</element>
<expression label="text5" exp=""D(" + (LaTeX[x(D)]) + "," + (LaTeX[y(D)]) + ")"" />
<element type="text" label="text5">
<show object="true" label="true" ev="8"/>
<condition showObject="A ≠ 1"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<algebra labelVisible="false"/> <isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="16" style="0"/>
<absoluteScreenLocation x="692" y="149"/>
</element>
</construction>
</geogebra>