Upward-Opening Parabola
2016-03-04 01:02
This applet demonstrates the geometric properties and construction of an upward-opening parabola. Using points, lines, orthogonal lines, and parabola tools, it illustrates the construction process involving the vertex, focus, and related segments. It is suitable for learning quadratic function graphs, coordinate geometry, and the basic properties of conic sections.
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.209.0" id="c9fe1346-fccb-440e-8fc4-6b01414cdf1a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1384" height="738" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.13439306358381503" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="333,85,700,550" />
<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 | 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="true" location="1" size="1368" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="186" window="100,100,250,400" />
<view id="16" visible="false" inframe="true" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="false" 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" />
<view id="512" visible="false" inframe="false" stylebar="false" location="1" size="300" 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 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6" position="1" help="true" />
<input show="true" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1368" height="523"/>
<coordSystem xZero="624.1013530765865" yZero="261.1673729394907" scale="9.673914923761844" yscale="9.673914923761851"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="13" 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="3" grid="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false" tickDistance="1.0"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false" tickDistance="1.0"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="1"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<command name="Point">
<input a0="yAxis"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="17.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="yAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="-7.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="B" a1="F"/>
<output a0="a"/>
</command>
<element type="line" label="a">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-24.0" y="0.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="B" a1="a"/>
<output a0="b"/>
</command>
<element type="line" label="b">
<show object="true" label="true"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="61" y="47"/>
<labelMode val="2"/>
<coords x="-0.0" y="-24.0" z="-168.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Parabola">
<input a0="F" a1="b"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="255" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="8" y="34"/>
<labelMode val="2"/>
<lineStyle thickness="1" type="20" typeHidden="1"/>
<eigenvectors x0="0.0" y0="1.0" z0="1.0" x1="1.0" y1="-0.0" z1="1.0"/>
<matrix A0="576.0" A1="0.0" A2="138240.0" A3="-0.0" A4="-0.0" A5="-13824.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="c" a1="a" a2="1"/>
<output a0="頂點"/>
</command>
<element type="point" label="頂點">
<show object="true" label="true"/>
<objColor r="0" g="0" b="204" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-2" y="32"/>
<labelMode val="1"/>
<coords x="0.0" y="5.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F" a1="頂點"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="true" label="true"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="21" y="-3"/>
<labelMode val="2"/>
<coords x="12.0" y="0.0" z="0.0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="頂點" a1="B"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="true"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="29" y="20"/>
<labelMode val="2"/>
<coords x="12.0" y="0.0" z="-0.0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="F" a1="d"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="12.0" z="-204.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="c" a1="f" a2="1"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-24.0" y="17.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="c" a1="f" a2="2"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="24.0" y="17.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="D"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="true"/>
<condition showObject="k"/>
<objColor r="204" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-76" y="-26"/>
<labelMode val="2"/>
<coords x="0.0" y="48.0" z="-816.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="false"/>
<condition showObject="l"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<trace val="true"/>
<animation step="1" speed="0.3" type="0" playing="false"/>
<coords x="6.456852791878144" y="5.8685614161663455" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="E" a1="b"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="24.0" y="-0.0" z="-154.96446700507545"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="b" a1="h"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="false"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3719.1472081218108" y="-4032.0" z="576.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F" a1="E"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="true"/>
<condition showObject="l"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="2"/>
<coords x="11.131438583833654" y="6.456852791878144" z="-109.76649746192845"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="G"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="true"/>
<condition showObject="l"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="2"/>
<coords x="12.868561416166346" y="0.0" z="-83.09040670742903"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""開口向上⇒c＞０ (x-h)²=4c(y-k)""/>
<element type="text" label="text1">
<show object="true" label="false"/>
<objColor r="0" g="51" b="153" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="115" y="129"/>
</element>
<element type="boolean" label="k">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="113" y="178"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="正焦弦長4lcl"/>
</element>
<element type="boolean" label="l">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="112" y="72"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="繪圖"/>
</element>
</construction>
</geogebra>