|00:01||Welcome to this tutorial on Conic Sections Parabola.|
|00:06||In this tutorial, we will learn how to use GeoGebra to:|
|00:11||Study standard equations and parts of a parabola|
|00:19||Here I am using:
Ubuntu Linux Operating System version 14.04, GeoGebra 5.0.388.0 hyphen d
|00:33||To follow this tutorial, you should have basic knowledge of
GeoGebra interface, Conic sections in geometry
|00:42||For relevant tutorials, please visit our website.|
A parabola is the locus of points equidistant from the fixed point called the focus.
|00:57||The points on the parabola are also equidistant from the fixed line called the directrix.|
|01:05||Observe the different features of the parabola in the image.|
|01:10||The Axis of Symmetry is perpendicular to the Directrix and passes through the Focus and Vertex.|
|01:19||Latus Rectum passes through the Focus and is perpendicular to the Axis of Symmetry.|
|01:27||Let us construct a parabola in GeoGebra.|
|01:31||I have already opened GeoGebra interface.|
|01:35||Click on Point tool and click in Graphics view.
This creates point A.
|01:44||Right-click on point A and select the Rename option.|
|01:49||In the New Name text box, type Focus instead of A and click OK.|
|01:57||This renames point A as Focus.|
|02:01||Click on Line tool and click on two places in Graphics view, below Focus.|
|02:10||This creates line AB.|
|02:14||Right-click on line AB and choose the Rename option.|
|02:20||In the New Name field, type directrix and click OK.|
|02:28||This renames line AB as the directrix.|
|02:33||Click on Perpendicular Line tool, then click on line AB.|
|02:41||Drag the cursor until the resulting line passes through Focus and click on Focus.|
|02:51||This draws a line perpendicular to line AB, passing through Focus.
|03:01||Right-click on this line perpendicular to line AB.|
|03:07||Choose the Rename option.|
|03:10||Type axis of symmetry in New Name field.
|03:19||Under Ellipse tool, click on Parabola tool.|
|03:25||Then click on Focus and the directrix.|
|03:31||This creates a parabola with its focus at Focus and with line AB as the directrix.|
|03:39||Under Point tool, click on Intersect tool.|
|03:44||Click on the parabola and axis of symmetry.|
|03:50||This creates point C at the intersection.|
|03:54||It is the vertex of the parabola.|
|03:58||Right-click on point C and choose the Rename option.|
|04:04||In the New Name field, type Vertex and click OK.|
|04:10||Click on Perpendicular Line tool and click on the axis of symmetry.|
|04:18||Drag the cursor until the line passes through the Focus and click on it.|
|04:24||This results in a line parallel to the directrix, passing through the Focus.|
|04:31||Under Point tool, click on Intersect tool.|
|04:35||Click on the parabola and the newly drawn line through Focus.|
|04:42||This creates points C and D.|
|04:46||Under Line tool, click on Segment tool and click on points C and D.|
|04:55||Resulting Segment CD is the latus rectum.|
|04:59||Right-click on Segment CD and choose the Rename option.|
|05:05||In the New Name field, type Latus Rectum and click OK.|
|05:13||Move the Latus label so you can see it properly.|
|05:18||Click and drag Graphics view to see the parabola properly.|
|05:24||In Algebra view, you can see the equation describing the parabola.|
|05:30||Drag boundary so you can see the equation properly.|
|05:35||Also, you can see the equations for the axis of symmetry, directrix and latus rectum.|
|05:43||Drag boundary so you can see Graphics view properly again.|
|05:49||Click in Graphics view and drag background.|
|05:54||Under Point tool, click on Intersect tool.|
|05:59||Click on axis of symmetry and directrix.|
|06:05||This creates point E.|
|06:08||Under Angle tool, click on Distance or Length tool.|
|06:13||Click on Focus and Vertex.|
|06:18||Note the distance of FocusVertex appearing in Graphics view.|
|06:23||Click on Vertex and point E.|
|06:27||Note the distance of Vertex E appearing in Graphics view.
Both these distances are equal.
|06:36||Let us look at the general equations of parabolas.|
|06:41||I have opened a new GeoGebra window.|
|06:45||In input bar, type x minus a in parentheses caret 2 equals 4 space p space y minus b in parentheses.|
|07:02||To type caret symbol, hold Shift key down and press 6.|
|07:09||Note that the spaces denote multiplication.
|07:16||Create Sliders window pops up asking if you want to create sliders for a, b and p.|
|07:24||Click on Create Sliders.|
|07:27||Sliders are created for a, p and b.|
|07:32||The default setting for all three coefficients is 1.|
|07:36||A parabola opening upwards appears in Graphics view.|
|07:41||a comma b correspond to the co-ordinates of the vertex.|
|07:47||Double click on the parabola, click on Object Properties and then on Color tab.|
|07:56||Select red and close the Preferences box.|
|08:00||The parabola and its equation appear red in the Graphics and Algebra views.|
|08:08||Move boundary so you can see the equation properly.|
|08:13||Right click on slider a and check Animation On option.|
|08:19||Note the effects on the horizontal movement of the red parabola.|
|08:24||Right-click on slider a and uncheck Animation On option.|
|08:30||Right-click on slider p and check Animation On option.|
|08:35||Note the effects on the shape and orientation of the parabola.|
|08:40||Right-click on slider p and uncheck Animation On option.|
|08:46||Right-click on slider b and check Animation On option.|
|08:52||Note the effects on the vertical movement of the parabola.|
|08:57||Right-click on slider b and uncheck Animation On option.|
|09:03||Note that when a, p and b are equal to 1, the red parabola c is described by equation c.|
|09:16||Click on parabola c in Graphics view and note highlighting of equation c in Algebra view.|
|09:25||Equation c is given by x squared minus 2x minus 4y equals minus 5.|
|09:33||In input bar, type Focus c in parentheses.
|09:41||Focus is drawn at point A in Graphics view.|
|09:46||The coordinates of Focus of parabola c, which is point A, appear in Algebra view.|
|09:53||In input bar, type Vertex c in parentheses.
|10:01||Vertex is drawn at point B in Graphics view.|
|10:05||The coordinates of Vertex of parabola c, which is point B, appear in Algebra view.|
|10:12||In input bar, type Directrix c in parentheses.
|10:21||Directrix appears as a line along x axis in Graphics view.|
|10:26||The equation for the Directrix of parabola c, y equals 0, appears in Algebra view.|
|10:34||Double click on Directrix in Graphics view.|
|10:38||In the Redefine text box, click on Object Properties, then the Color tab.|
|10:45||In the left panel, note that the Directrix is highlighted.|
|10:51||Identify Focus and Vertex created for parabola c.|
|10:56||While pressing the Control key, click and highlight Focus and Vertex.|
|11:03||Click on red.|
|11:06||Close the Preferences box.|
|11:09||For parabola c, Focus, Vertex and Directrix and their coordinates and equation appear red.|
|11:19||Follow the earlier steps to construct parabola d.|
|11:24||Let us summarize.|
|11:26||In this tutorial, we have learnt how to use GeoGebra to:
Study the standard equations and parts of a parabola, Construct parabolas
|11:36||As an assignment:
Try these steps to construct parabolas with these foci and directrices.
|11:45||Find their equations.|
|11:47||As an assignment: Find the coordinates of the foci and length of the latus recti for these parabolas.|
|11:56||Also, find the equations of the axes of symmetry and directrices.|
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