Applications-of-GeoGebra/C2/Conic-Sections-Parabola/English-timed

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Time Narration
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
00:16 Construct parabolas.
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.
00:46 Parabola

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.


This line is the axis of symmetry.

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.

Click OK.

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.

Press Enter.

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.

Press Enter.

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.

Press Enter.

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.

Press Enter.

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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12:37 This is Vidhya Iyer from IIT Bombay, signing off.

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