三次函数及其图像性质
2020-11-17 08:59
本范例演示了三次函数的图像及其相关性质。通过GeoGebra的交点命令，展示了函数图像与坐标轴或其他几何对象的交点关系，帮助直观理解三次函数的根、极值点及图像的整体走势与特征。
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="5.0.613.0" app="classic" platform="w" id="205431cb-a7b6-45e3-9779-d83781773975" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1271" height="571" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.22974036191974823" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1,1,1" size="400" window="290,129,700,550" />
<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="2" visible="false" inframe="false" stylebar="false" location="3" size="292" tab="ALGEBRA" 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="true" 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="1" visible="true" inframe="false" stylebar="true" location="" size="569" window="100,100,600,400" />
</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="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="2"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1268" height="568"/>
<coordSystem xZero="399.58914515119864" yZero="292.2566766048898" scale="25.868510011997184" yscale="25.86851001199711"/>
<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="false"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="1"/>
<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="numeric" label="a">
<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="200" x="733" y="338" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.5" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="p">
<value val="0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-37" y="6"/>
<labelMode val="1"/>
<slider min="-10" max="10" absoluteScreenLocation="true" width="200" x="732" y="374" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.5" 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="-4" max="4" absoluteScreenLocation="true" width="200" x="733" y="414" fixed="true" 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="0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-10" max="10" absoluteScreenLocation="true" width="200" x="734" y="453" fixed="true" 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 * x^(3)) + (p * x)" />
<element type="function" label="f">
<show object="true" label="true" ev="4"/>
<condition showObject="c"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-12" y="-47"/>
<labelMode val="1"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<caption val="f(x)=a x³ + p x"/>
</element>
<expression label="g" exp="g(x) = (a * (x + h)^(3)) + (p * (x + h)) + k" />
<element type="function" label="g">
<show object="true" label="true" ev="4"/>
<condition showObject="d"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-4" y="-80"/>
<labelMode val="1"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<caption val="g(x)=a (x + h)³ + p (x + h) + k"/>
</element>
<element type="boolean" label="c">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="56" y="5"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="顯示 f(x)"/>
</element>
<expression label="l" exp="l(x) = (p * (x + h)) + k" />
<element type="function" label="l">
<show object="true" label="true" ev="4"/>
<condition showObject="b"/>
<objColor r="102" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-3" y="39"/>
<labelMode val="1"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<element type="boolean" label="b">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="56" y="65"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="顯示『一次近似』"/>
</element>
<command name="Intersect">
<input a0="g" a1="l" a2="1"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<condition showObject="d"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="2" y="1"/>
<labelMode val="1"/>
<coords x="-2" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="boolean" label="d">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="56" y="35"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="顯示 g(x)"/>
</element>
<expression label="text1" exp=""\underline{三次函數}:\\ \\ 
(1)改變a值，\;觀察圖形變化\\
(2)改變a,p值，\;觀察a和p同號、\;異號圖形與x軸的相交狀況\\
(3)顯示g(x)\;改變h,k值，\;以對稱中心A=" + (LaTeX[A]) + "\\
\;\;\;\;\;來觀察f(x)與g(x)之平移關係\\
(4)顯示一次近似，\;觀察g(x)與l(x)在A點附近的貼合狀況"" />
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="224" g="224" b="224" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="551" y="39"/>
</element>
</construction>
</geogebra>