Properties of Parabola

2020-03-10 00:47

This GeoGebra example demonstrates the core geometric properties of parabolas. By constructing the Focus and Directrix, it illustrates standard parabola forms opening in different directions (up, down, left, right). The Vector tool is used to analyze distance relationships between the focus and points on the parabola, while TableText displays geometric quantities corresponding to parameters p, p/2

#coordinate-geometry #vector #locus #geometry-construction

In collections Secondary Vocational Mathematics

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	<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="1338" height="565"/>
	<coordSystem xZero="270.0908658463004" yZero="273.6029277477111" scale="8.830108317772192" yscale="8.83010760651431"/>
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	<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="2" showNumbers="false"/>
	<axis id="1" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
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<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="numeric" label="p">
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	<show object="true" label="true"/>
	<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
	<layer val="0"/>
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	<slider min="1" max="20" absoluteScreenLocation="true" width="200" x="611" y="163" fixed="false" horizontal="true" showAlgebra="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
	<caption val="$\bold \Large %n=%v$"/>
</element>
<expression label="halfp" exp="p / 2" />
<element type="numeric" label="halfp">
	<value val="6"/>
	<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="twop" exp="(2 * p)" />
<element type="numeric" label="twop">
	<value val="24"/>
	<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<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"/>
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<element type="point" label="dDR">
	<show object="true" label="false" ev="4"/>
	<objColor r="77" g="77" b="255" alpha="0"/>
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	<labelMode val="0"/>
	<animation step="0.1" speed="1" type="1" playing="false"/>
	<coords x="31.47290204745927" y="-30.508922909789277" 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="35" g="80" b="75" alpha="1"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<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="96" g="96" b="96" alpha="0"/>
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	<coordStyle style="cartesian"/>
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</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="96" g="96" b="96" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
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	<coordStyle style="cartesian"/>
	<startPoint  x="0" y="-30.508922909789277" z="1"/>
</element>
<command name="Text">
	<input a0="&quot;y&quot;" a1="(0, y(dUL) - 1)" a2="true" a3="true"/>
	<output a0="text2"/>
</command>
<element type="text" label="text2">
	<show object="true" label="true" 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"/>
	<startPoint  x="0" y="27.607004467460413" z="1"/>
</element>
<command name="Text">
	<input a0="&quot;x&quot;" a1="(x(dDR) - 1.5, 0)" a2="true" a3="true"/>
	<output a0="text1"/>
</command>
<element type="text" label="text1">
	<show object="true" label="true" ev="8"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
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	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="1.4" size="5" style="0"/>
	<startPoint  x="29.97290204745927" y="0" z="1"/>
</element>
<expression label="text3" exp="&quot;抛物线的性质&quot;"/>
<element type="text" label="text3">
	<show object="true" label="true" ev="40"/>
	<objColor r="255" g="0" b="0" alpha="0"/>
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	<font serif="false" sizeM="2" size="12" style="1"/>
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<expression label="pr" exp="y^(2) = (2 * (p * x))" type="conic" />
<element type="conic" label="pr">
	<show object="true" label="true" ev="4"/>
	<condition showObject="sr"/>
	<objColor r="255" g="0" b="0" alpha="0"/>
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	<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
	<eigenvectors  x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
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	<eqnStyle style="specific"/>
</element>
<expression label="pl" exp="y^(2) = (-2 * (p * x))" type="conic" />
<element type="conic" label="pl">
	<show object="true" label="true" ev="4"/>
	<condition showObject="sl"/>
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	<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
	<eigenvectors  x0="-1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
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	<eqnStyle style="specific"/>
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<expression label="pu" exp="x^(2) = (2 * (p * y))" type="conic" />
<element type="conic" label="pu">
	<show object="true" label="true" ev="4"/>
	<condition showObject="su"/>
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<expression label="pd" exp="x^(2) = (-2 * (p * y))" type="conic" />
<element type="conic" label="pd">
	<show object="true" label="true" ev="4"/>
	<condition showObject="sd"/>
	<objColor r="255" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
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	<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
	<eigenvectors  x0="0" y0="-1" z0="1.0" x1="-1" y1="0" z1="1.0"/>
	<matrix A0="1" A1="0" A2="0" A3="0" A4="0" A5="12"/>
	<eqnStyle style="specific"/>
</element>
<command name="Focus">
	<input a0="pr"/>
	<output a0="Fr" a1=""/>
</command>
<element type="point" label="Fr">
	<show object="true" label="true" ev="4"/>
	<condition showObject="sr"/>
	<objColor r="255" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="3"/>
	<coords x="6" y="0" z="1"/>
	<pointSize val="4"/>
	<pointStyle val="0"/>
	<caption val="$\Large F%v%$"/>
</element>
<command name="Focus">
	<input a0="pl"/>
	<output a0="Fl" a1=""/>
</command>
<element type="point" label="Fl">
	<show object="true" label="true" ev="4"/>
	<condition showObject="sl"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
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	<labelMode val="3"/>
	<coords x="-6" y="0" z="1"/>
	<pointSize val="4"/>
	<pointStyle val="0"/>
	<caption val="$\Large F%v%$"/>
</element>
<command name="Focus">
	<input a0="pu"/>
	<output a0="Fu" a1=""/>
</command>
<element type="point" label="Fu">
	<show object="true" label="true" ev="4"/>
	<condition showObject="su"/>
	<objColor r="128" g="0" b="128" alpha="0"/>
	<layer val="0"/>
	<labelMode val="3"/>
	<coords x="0" y="6" z="1"/>
	<pointSize val="4"/>
	<pointStyle val="0"/>
	<caption val="$\Large F%v%$"/>
</element>
<command name="Focus">
	<input a0="pd"/>
	<output a0="Fd" a1=""/>
</command>
<element type="point" label="Fd">
	<show object="true" label="true" ev="4"/>
	<condition showObject="sd"/>
	<objColor r="255" g="0" b="255" alpha="0"/>
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	<coords x="0" y="-6" z="1"/>
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	<pointStyle val="0"/>
	<caption val="$\Large F%v%$"/>
</element>
<command name="Directrix">
	<input a0="pr"/>
	<output a0="zr" a1=""/>
</command>
<element type="line" label="zr">
	<show object="true" label="true" ev="4"/>
	<condition showObject="sr"/>
	<objColor r="255" g="0" b="0" alpha="0"/>
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	<coords x="-1" y="0" z="-6"/>
	<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
	<eqnStyle style="implicit"/>
	<caption val="$\Large %v$"/>
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<command name="Directrix">
	<input a0="pl"/>
	<output a0="zl" a1=""/>
</command>
<element type="line" label="zl">
	<show object="true" label="true" ev="4"/>
	<condition showObject="sl"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
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	<coords x="-1" y="0" z="6"/>
	<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
	<eqnStyle style="implicit"/>
	<caption val="$\Large %v$"/>
</element>
<command name="Directrix">
	<input a0="pu"/>
	<output a0="zu" a1=""/>
</command>
<element type="line" label="zu">
	<show object="true" label="true" ev="4"/>
	<condition showObject="su"/>
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	<caption val="$\Large %v$"/>
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<command name="Directrix">
	<input a0="pd"/>
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<element type="line" label="zd">
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	<condition showObject="sd"/>
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	<coords x="0" y="-1" z="6"/>
	<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
	<eqnStyle style="implicit"/>
	<caption val="$\Large %v$"/>
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<element type="boolean" label="sr">
	<value val="false"/>
	<show object="true" label="true"/>
	<objColor r="255" g="255" b="255" alpha="0"/>
	<layer val="2"/>
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	<labelMode val="3"/>
	<checkbox fixed="true"/>
	<caption val="向右"/>
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<element type="boolean" label="sl">
	<value val="false"/>
	<show object="true" label="true"/>
	<objColor r="255" g="255" b="255" alpha="0"/>
	<layer val="2"/>
	<labelOffset x="750" y="78"/>
	<labelMode val="3"/>
	<checkbox fixed="true"/>
	<caption val="向左"/>
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<element type="boolean" label="su">
	<value val="true"/>
	<show object="true" label="true"/>
	<objColor r="255" g="255" b="255" alpha="0"/>
	<layer val="2"/>
	<labelOffset x="591" y="79"/>
	<labelMode val="0"/>
	<checkbox fixed="true"/>
	<caption val="向上"/>
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<expression label="xlr" exp="{((&quot;\red{y^2=&quot; * twop) * &quot;x}&quot;), &quot;\red{向右}&quot;, &quot;\red{x轴}&quot;, ((&quot;\red{(&quot; * halfp) * &quot;,0)}&quot;), ((&quot;\red{x=-&quot; * halfp) * &quot;}&quot;)}" />
<element type="list" label="xlr">
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	<lineStyle thickness="5" type="0" typeHidden="1"/>
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<element type="boolean" label="sd">
	<value val="false"/>
	<show object="true" label="true"/>
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	<labelOffset x="669" y="79"/>
	<labelMode val="0"/>
	<checkbox fixed="true"/>
	<caption val="向下"/>
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<expression label="xl标题" exp="{&quot;方程&quot;, &quot;开口方向&quot;, &quot;对称轴&quot;, &quot;焦点坐标&quot;, &quot;准线方程&quot;}"/>
<element type="list" label="xl标题">
	<show object="false" label="true" ev="4"/>
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	<symbolic val="true" />
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<expression label="xll" exp="{((&quot;\blue{y^2=-&quot; * twop) * &quot;x}&quot;), &quot;\blue{向左}&quot;, &quot;\blue{x轴}&quot;, ((&quot;\blue{(-&quot; * halfp) * &quot;,0)}&quot;), ((&quot;\blue{x=&quot; * halfp) * &quot;}&quot;)}" />
<element type="list" label="xll">
	<show object="false" label="true" ev="4"/>
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	<lineStyle thickness="5" type="0" typeHidden="1"/>
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	<angleStyle val="0"/>
	<symbolic val="true" />
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<expression label="xlu" exp="{((&quot;\purple{x^2=&quot; * twop) * &quot;y}&quot;), &quot;\purple{向上}&quot;, &quot;\purple{y轴}&quot;, ((&quot;\purple{(0,&quot; * halfp) * &quot;)}&quot;), ((&quot;\purple{y=-&quot; * halfp) * &quot;}&quot;)}" />
<element type="list" label="xlu">
	<show object="false" label="true" ev="4"/>
	<objColor r="0" g="100" b="0" alpha="0"/>
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	<lineStyle thickness="5" type="0" typeHidden="1"/>
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	<symbolic val="true" />
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<expression label="xld" exp="{((&quot;\magenta{x^2=-&quot; * twop) * &quot;y}&quot;), &quot;\magenta{向下}&quot;, &quot;\magenta{y轴}&quot;, ((&quot;\magenta{(0,-&quot; * halfp) * &quot;)}&quot;), ((&quot;\magenta{y=&quot; * halfp) * &quot;}&quot;)}" />
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<command name="TableText">
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