The Three Forms of Quadratic Functions
2014-08-27 22:00
This interactive applet allows students to explore and understand the three different forms of quadratic functions: standard form, vertex form, and intercept form. By manipulating parameters dynamically, users can visually observe how changes in the equations affect the parabola's shape, vertex position, and x-intercepts, bridging the gap between algebraic expressions and geometric representations
In collections
初中函数
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<command name="LineBisector">
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<eqnStyle style="implicit"/>
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<command name="Intersect">
<input a0="h_1" a1="a"/>
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</command>
<element type="point" label="E">
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<condition showObject="i ≠ 0"/>
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<checkbox fixed="true"/>
<caption val="Factored form"/>
</element>
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<condition showObject="l ≠ 0"/>
<objColor r="102" g="0" b="102" alpha="0.1"/>
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<slider min="-3" max="3" absoluteScreenLocation="true" width="200" x="25" y="442" fixed="true" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
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<element type="numeric" label="h">
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<condition showObject="l ≠ 0"/>
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<slider min="-10" max="10" absoluteScreenLocation="true" width="200" x="25" y="487" fixed="true" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="k">
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<condition showObject="l ≠ 0"/>
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<slider min="-8" max="8" absoluteScreenLocation="true" width="200" x="27" y="535" fixed="true" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<animation step="0.5" speed="1" type="0" playing="false"/>
</element>
<expression label="r" exp="y = (a_V * (x - h)^(2)) + k" type="conic" />
<element type="conic" label="r">
<show object="true" label="false" ev="7"/>
<condition showObject="l ≠ 0"/>
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<lineStyle thickness="6" type="0" typeHidden="1"/>
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</element>
<command name="Intersect">
<input a0="r" a1="xAxis"/>
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</command>
<element type="point" label="F">
<show object="true" label="true" ev="7"/>
<condition showObject="l ≠ 0"/>
<objColor r="102" g="0" b="102" alpha="0"/>
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<element type="point" label="G">
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<condition showObject="l ≠ 0"/>
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<command name="LineBisector">
<input a0="F" a1="G"/>
<output a0="j"/>
</command>
<element type="line" label="j">
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<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
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<element type="boolean" label="l">
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<show object="true" label="true"/>
<objColor r="102" g="0" b="102" alpha="0"/>
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<labelOffset x="19" y="376"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="Vertex form"/>
</element>
<command name="Vertex">
<input a0="r"/>
<output a0="J" a1="" a2="" a3=""/>
</command>
<element type="point" label="J">
<show object="true" label="true" ev="7"/>
<condition showObject="l ≠ 0"/>
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<command name="Vertex">
<input a0="d"/>
<output a0="B" a1="" a2="" a3=""/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="7"/>
<condition showObject="f ≠ 0"/>
<objColor r="153" g="0" b="51" alpha="0"/>
<layer val="0"/>
<labelMode val="2"/>
<coords x="0" y="0" z="1"/>
<pointSize val="6"/>
<pointStyle val="1"/>
</element>
<expression label="text3" exp=""y =" + a_V + "(x - " + h + ")² +" + k + """ />
<element type="text" label="text3">
<show object="true" label="false" ev="8"/>
<condition showObject="l ≠ 0"/>
<objColor r="102" g="0" b="102" alpha="0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
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<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1" size="0" style="1"/>
<absoluteScreenLocation x="245" y="453"/>
</element>
<expression label="text2" exp=""y=" + a_F + "(x - " + p + ")(x - " + q + ")"" />
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<condition showObject="i ≠ 0"/>
<objColor r="0" g="0" b="204" alpha="0"/>
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<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1" size="0" style="1"/>
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</element>
<expression label="d_1" exp="y = (a_E * x^(2)) + (b * x) + c" type="conic" />
<element type="conic" label="d_1">
<show object="false" label="false" ev="4"/>
<objColor r="204" g="0" b="51" alpha="0"/>
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<lineStyle thickness="6" type="0" typeHidden="1"/>
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<expression label="e_1" exp="b^(2) - ((4 * a_E) * c)" />
<element type="numeric" label="e_1">
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<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Vertex">
<input a0="d_1"/>
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</command>
<element type="point" label="B_1">
<show object="false" label="true" ev="4"/>
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<layer val="0"/>
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</element>
<expression label="text1_1" exp=""y = ax² +bx +c""/>
<element type="text" label="text1_1">
<show object="true" label="true" ev="8"/>
<objColor r="204" g="0" b="51" alpha="0"/>
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<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1" size="0" style="1"/>
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</element>
<expression label="h_2" exp="y = ((a_F * (x - p)) * (x - q))" type="conic" />
<element type="conic" label="h_2">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
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<lineStyle thickness="6" type="0" typeHidden="1"/>
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<eqnStyle style="explicit"/>
</element>
<expression label="text2_1" exp=""y=a(x - p)(x - q)""/>
<element type="text" label="text2_1">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="204" alpha="0"/>
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<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1" size="0" style="1"/>
<absoluteScreenLocation x="247" y="220"/>
</element>
<expression label="r_1" exp="y = (a_V * (x - h)^(2)) + k" type="conic" />
<element type="conic" label="r_1">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="0" b="102" alpha="0"/>
<layer val="0"/>
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<lineStyle thickness="6" type="0" typeHidden="1"/>
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<eqnStyle style="explicit"/>
</element>
<expression label="text3_1" exp=""y =a(x - h)² +k""/>
<element type="text" label="text3_1">
<show object="true" label="false" ev="8"/>
<objColor r="102" g="0" b="102" alpha="0"/>
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<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1" size="0" style="1"/>
<absoluteScreenLocation x="244" y="394"/>
</element>
</construction>
</geogebra>