Opening, Axis of Symmetry, and Vertex of a Parabola
2020-01-27 14:41
This applet demonstrates the fundamental geometric properties of a quadratic function (parabola), including the direction of opening, the axis of symmetry, and the vertex coordinates. By dynamically adjusting parameters, users can visually observe how these features change with the function's equation, aiding in a deeper understanding of the graphical properties of quadratic functions.
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.570.0" app="classic" platform="d" id="fb460cdf-8877-42cf-ba07-98082fa7522e" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1942" height="1042" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.1431513903192585" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="554,237,1046,686" />
<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 , 22 , 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="true" location="1,1,1" size="500" window="100,100,600,400" />
<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="true" 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="true" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1920" window="212,169,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="278" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="true" 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="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="false" 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="false" cmd="false" top="false" />
<dockBar show="false" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1920" height="914"/>
<coordSystem xZero="925.0360580000003" yZero="473.1653760000016" scale="80.52549999999992" yscale="80.52550000000045"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="0" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="0" g="0" b="0"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="1"/>
</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="大小">
<value val="5.0"/>
<show object="false" label="true"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="2"/>
<labelOffset x="-41" y="6"/>
<labelMode val="1"/>
<slider min="1" max="10" absoluteScreenLocation="true" width="80.0" x="1310.0" y="63.0" 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="O" exp="(0, 0)" type="point"/>
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="0" y="27"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="text1" exp=""x""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="24" style="1"/>
<startPoint exp="(大小 - y軸位移, 0)"/>
<labelOffset x="-5" y="-29"/>
</element>
<expression label="text1_1" exp=""y""/>
<element type="text" label="text1_1">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="24" style="1"/>
<startPoint exp="(0, 大小 - x軸位移)"/>
<labelOffset x="-24" y="-55"/>
</element>
<element type="numeric" label="y軸位移">
<value val="0.0"/>
<show object="false" label="true"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="2"/>
<labelOffset x="-39" y="3"/>
<labelMode val="1"/>
<slider min="(-大小)" max="大小" absoluteScreenLocation="true" width="80.0" x="1315.0" y="179.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Sequence">
<input a0="Segment[((-大小) - y軸位移 + k, 0.1), ((-大小) - y軸位移 + k, -0.1)]" a1="k" a2="1" a3="(2 * 大小) - 1"/>
<output a0="x軸刻度"/>
</command>
<element type="list" label="x軸刻度">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="8" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="x軸位移">
<value val="0.0"/>
<show object="false" label="true"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="2"/>
<labelOffset x="-69" y="6"/>
<labelMode val="1"/>
<slider min="(-大小)" max="大小" absoluteScreenLocation="true" width="80.0" x="1313.0" y="118.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Vector">
<input a0="((-大小) - y軸位移, 0)" a1="(大小 - y軸位移, 0)"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10.0" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint x="-5.0" y="0.0" z="1.0"/>
</element>
<command name="Vector">
<input a0="(0, (-大小) - x軸位移)" a1="(0, 大小 - x軸位移)"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="10.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint x="0.0" y="-5.0" z="1.0"/>
</element>
<command name="Sequence">
<input a0="Segment[(-0.1, (-大小) - x軸位移 + k), (0.1, (-大小) - x軸位移 + k)]" a1="k" a2="1" a3="(2 * 大小) - 1"/>
<output a0="y軸刻度"/>
</command>
<element type="list" label="y軸刻度">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="8" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Segment[((-大小) - y軸位移, t), (大小 - y軸位移, t)]" a1="t" a2="(-大小) - x軸位移" a3="大小 - x軸位移"/>
<output a0="y軸格線"/>
</command>
<element type="list" label="y軸格線">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Segment[(t, (-大小) - x軸位移), (t, 大小 - x軸位移)]" a1="t" a2="(-大小) - y軸位移" a3="大小 - y軸位移"/>
<output a0="x軸格線"/>
</command>
<element type="list" label="x軸格線">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="a">
<value val="1.0"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="2"/>
<labelOffset x="-9" y="11"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="300.0" x="72.0" y="403.0" fixed="true" horizontal="true" showAlgebra="true"/>
<pointSize val="20.0"/>
<lineStyle thickness="40" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="h">
<value val="-1.0"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="2"/>
<labelOffset x="-11" y="6"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="300.0" x="73.0" y="483.0" fixed="true" horizontal="true" showAlgebra="true"/>
<pointSize val="20.0"/>
<lineStyle thickness="40" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="k">
<value val="-2.0"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="2"/>
<labelOffset x="-13" y="11"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="300.0" x="73.0" y="566.0" fixed="true" horizontal="true" showAlgebra="true"/>
<pointSize val="20.0"/>
<lineStyle thickness="40" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="lr">
<value val="0.0"/>
<show object="false" label="true"/>
<objColor r="0" g="255" b="255" alpha="0.10000000149011612"/>
<layer val="2"/>
<labelOffset x="-10" y="4"/>
<labelMode val="1"/>
<slider min="0" max="abs(2)" absoluteScreenLocation="true" width="100.0" x="82.0" y="637.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="5" type="3" playing="false"/>
</element>
<element type="numeric" label="ud">
<value val="0.0"/>
<show object="false" label="true"/>
<objColor r="0" g="255" b="255" alpha="0.10000000149011612"/>
<layer val="2"/>
<labelOffset x="-18" y="6"/>
<labelMode val="1"/>
<slider min="0" max="abs(3)" absoluteScreenLocation="true" width="100.0" x="81.0" y="670.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="If[lr ≟ abs(2), 5, 0]" type="3" playing="false"/>
</element>
<expression label="f" exp="f(x) = If[(-大小) - y軸位移 < x < 大小 - y軸位移, (a * (x - h)^(2)) + k]" />
<element type="function" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
</element>
<element type="image" label="pic1">
<file name="2efd0399102971428df2b61de4c50d89/P109-對稱點.png"/>
<inBackground val="false"/>
<absoluteScreenLocation x="37" y="602"/> <show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="2"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label="V" exp="(h, k)" type="point" />
<element type="point" label="V">
<show object="true" label="false"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="-1.0" y="-2.0" z="1.0"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<expression label="text2" exp=""V" + (LaTeX[V]) + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="10" style="1"/>
<startPoint exp="V"/>
<labelOffset x="-64" y="6"/>
</element>
<expression label="eq1" exp="x = h" type="line" />
<element type="line" label="eq1">
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="1.0" y="0.0" z="1.0"/>
<lineStyle thickness="8" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<expression label="text5" exp=""" + (LaTeX[eq1]) + """ />
<element type="text" label="text5">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="10" style="1"/>
<startPoint exp="V + (-1, 1)"/>
<labelOffset x="4" y="-58"/>
</element>
<expression label="text3" exp=""y=a(x-h)^2+k\\
y=" + (LaTeX[If[a > 1, a + "", a < -1, a + "", a ≟ 0, "0", a ≟ 1, "", a ≟ -1, "-"]]) + "\green{" + (LaTeX[LaTeX[If[h ≟ 0, "", "("]]]) + "
x" + (LaTeX[If[h < 0, "+" + (abs(h)), h ≟ 0, "", h > 0, "-" + h]]) + "" + (LaTeX[If[h ≟ 0, "", ")"]]) + "}^2" + (LaTeX[If[k ≟ 0, "", k < 0, k + "", k > 0, "+" + k]]) + "\\
\\
\\\yellow{" + (LaTeX[If[a ≟ 0, "", "頂點(" + h + "," + k + ")"]]) + "}\\
\\\cyan{" + (LaTeX[If[a ≟ 0, "", a < 0, "開口向下", "開口向上"]]) + "}"" />
<element type="text" label="text3">
<show object="true" label="false" ev="8"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="24" style="1"/>
<startPoint exp="(大小, 大小)"/>
<labelOffset x="50" y="10"/>
</element>
<element type="image" label="pic2">
<file name="fab93c71e805c62dd7a3e1e1658e0ef0/拋物線的開口、對稱軸、頂點.png"/>
<inBackground val="false"/>
<absoluteScreenLocation x="27" y="215"/> <show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
</element>
</construction>
</geogebra>