Difference between revisions of "Applications-of-GeoGebra/C2/Conic-Sections-Parabola/English"

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* Construct parabolas.
 
* Construct parabolas.
 
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||'''Slide Number 3'''
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|| '''Slide Number 3'''
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'''Pre-requisites'''
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||To follow this '''tutorial''', you should have basic knowledge of
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'''GeoGebra''' interface
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'''Conic sections''' in geometry
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|| '''Slide Number 4'''
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'''System Requirement'''
 
'''System Requirement'''
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'''GeoGebra 5.0.388.0-d'''
 
'''GeoGebra 5.0.388.0-d'''
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||'''Slide Number 4'''
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'''Pre-requisites'''
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www.spoken-tutorial.org
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||To follow this '''tutorial''', you should have basic knowledge of
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'''GeoGebra''' interface
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'''Conic sections''' in geometry
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For relevant tutorials, please visit our website. 
 
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||'''Slide Number 5'''
 
||'''Slide Number 5'''
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The points on the parabola are also equidistant from the fixed line called the '''directrix'''.  
 
The points on the parabola are also equidistant from the fixed line called the '''directrix'''.  
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||A parabola is the '''locus''' of points equidistant from the fixed point called the focus.
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||Parabola
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A parabola is the '''locus''' of points equidistant from the fixed point called the focus.
  
 
The points on the parabola are also equidistant from the fixed line called the '''directrix'''.  
 
The points on the parabola are also equidistant from the fixed line called the '''directrix'''.  
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I have already opened '''GeoGebra''' interface.
 
I have already opened '''GeoGebra''' interface.
 
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||Click on '''Point''' tool and click in '''Graphics''' view.  
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||Click on '''Point''' tool >> click in '''Graphics''' view.  
  
 
Point to point '''A'''.  
 
Point to point '''A'''.  
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Type '''axis of symmetry''' in '''New Name''' field.  
 
Type '''axis of symmetry''' in '''New Name''' field.  
  
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Click '''OK'''
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Click '''OK'''.
 
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||Click on '''Parabola''' tool under '''Ellipse''' tool.
 
||Click on '''Parabola''' tool under '''Ellipse''' tool.
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Press '''Enter'''.  
 
Press '''Enter'''.  
 
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||Point to '''Create Sliders''' window
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||Point to '''Create Sliders''' window.
 
||'''Create Sliders''' window pops up asking if you want to create '''sliders''' for '''a, b''' and '''p'''.
 
||'''Create Sliders''' window pops up asking if you want to create '''sliders''' for '''a, b''' and '''p'''.
 
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'''Assignment'''
 
'''Assignment'''
  
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Find the coordinates of the '''foci''' and length of '''latus recti''' for these parabolas.  
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Find the coordinates of the '''foci''' >> length of '''latus recti''' for these parabolas.  
  
 
Also, find the equations of the '''axes of symmetry''' and '''directrices'''.  
 
Also, find the equations of the '''axes of symmetry''' and '''directrices'''.  

Latest revision as of 15:50, 24 July 2018

Visual Cue Narration
Slide Number 1

Title Slide

Welcome to this tutorial on Conic Sections – Parabola.
Slide Number 2

Learning Objectives

In this tutorial, we will learn how to use GeoGebra to:
  • Study standard equations and parts of a parabola
  • Construct parabolas.
Slide Number 3

System Requirement

Here I am using:

Ubuntu Linux Operating System version 14.04

GeoGebra 5.0.388.0-d

Slide Number 4

Pre-requisites


www.spoken-tutorial.org

To follow this tutorial, you should have basic knowledge of

GeoGebra interface

Conic sections in geometry

For relevant tutorials, please visit our website.

Slide Number 5

Parabola

A parabola is the locus of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the directrix.

Parabola

A parabola is the locus of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the directrix.

Observe the different features of the parabola in the image.

The Axis of Symmetry is perpendicular to the Directrix and passes through the Focus and Vertex.

Latus Rectum passes through the Focus and is perpendicular to the Axis of Symmetry.

Show the GeoGebra window. Let us construct a parabola in GeoGebra.

I have already opened GeoGebra interface.

Click on Point tool >> click in Graphics view.

Point to point A.

Click on Point tool and click in Graphics view.

This creates point A.

Right-click on point A >> select the Rename option. Right-click on point A and select the Rename option.
In New Name text box, type Focus instead of A >> click OK.

Point to Focus.

In the New Name text box, type Focus instead of A and click OK.

This renames point A as Focus.

Click on Line tool >> click in two places in Graphics view below Focus.

Point to line AB.

Click on Line tool and click on two places in Graphics view, below Focus.

This creates line AB.

Right-click on line AB >> choose Rename option. Right-click on line AB and choose the Rename option.
Type directrix in New Name field >> click OK.

Point to directrix.

In the New Name field, type directrix and click OK.

This renames line AB as the directrix.

Click on Perpendicular Line tool >> click on line AB. Click on Perpendicular Line tool, then click on line AB.
Drag the cursor until Focus >> click on point A. Drag the cursor until the resulting line passes through Focus and click on Focus.
Point to the perpendicular line through Focus.

Point to axis of symmetry.

This draws a line perpendicular to line AB, passing through Focus.


This line is the axis of symmetry.

Right-click on this line perpendicular to line AB.

Choose the Rename option.

Type axis of symmetry in New Name field >> click OK.

Right-click on this line perpendicular to line AB.

Choose the Rename option.

Type axis of symmetry in New Name field.

Click OK.

Click on Parabola tool under Ellipse tool.

Click on Focus and line AB (directrix).

Under Ellipse tool, click on Parabola tool.

Then click on Focus and the directrix.

Point to the parabola. This creates a parabola with its focus at Focus and with line AB as the directrix.
Click on Intersect tool. >> Click on the parabola and axis of symmetry. Under Point tool, click on Intersect tool.

Click on the parabola and axis of symmetry.

Point to point C. This creates point C at the intersection.

It is the vertex of the parabola.

Right-click on point C >> choose the Rename option. Right-click on point C and choose the Rename option.
Type Vertex in the New Name field >> click OK. In the New Name field, type Vertex and click OK.
Click on Perpendicular Line tool >> click on the axis of symmetry. Click on Perpendicular Line tool and click on the axis of symmetry.
Drag the cursor until the line passes through point A (Focus) >> click on point A. Drag the cursor until the line passes through the Focus and click on it.
Point to the parallel line. This results in a line parallel to the directrix, passing through the Focus.
Click on Intersect tool under Point tool.


Click on the intersections of the parabola and the newly drawn line through Focus.

Under Point tool, click on Intersect tool.

Click on the parabola and the newly drawn line through Focus.

Point to points C and D. This creates points C and D.
Click on Segment tool under the Line tool >> click on points Cand D. Under Line tool, click on Segment tool and click on points C and D.
Point to Segment CD. Resulting Segment CD is the latus rectum.
Right-click on Segment CD >> choose the Rename option. Right-click on Segment CD and choose the Rename option.
Type Latus Rectum in the New Name field >> click OK button. In the New Name field, type Latus Rectum and click OK.
Move the Latus label so you can see it properly. Move the Latus label so you can see it properly.
Click >> drag Graphics view to see the parabola properly. Click and drag Graphics view to see the parabola properly.
Point to Algebra view.

Drag boundary so you can see equation properly.

In Algebra view, you can see the equation describing the parabola.

Drag boundary so you can see the equation properly.

Also, you can see the equations for the axis of symmetry, directrix and latus rectum.

Drag boundary so you can see Graphics view properly. Drag boundary so you can see Graphics view properly again.
Click in Graphics view and drag background. Click in Graphics view and drag background.
Click on Intersect tool under Point tool.

Click on the intersection of the axis of symmetry and the directrix.

Under Point tool, click on Intersect tool.

Click on axis of symmetry and directrix.

Point to point E. This creates point E.
Click on Distance or Length tool under Angle tool. Under Angle tool, click on Distance or Length tool.
Click on Focus >> Vertex. Click on Focus and Vertex.
Point to the distance of FocusVertex appearing in Graphics view. Note the distance of FocusVertex appearing in Graphics view.
Click on Vertex >> point E. Click on Vertex and point E.
Point to the distance of Vertex E appearing in Graphics view. Note the distance of Vertex E appearing in Graphics view.

Both these distances are equal.

Let us look at the general equations of parabolas.
Show the new GeoGebra window. I have opened a new GeoGebra window.
Type (x-a)^2=4 p (y-b) in input bar >> press Enter. In input bar, type x minus a in parentheses caret 2 equals 4 space p space y minus b in parentheses.

To type caret symbol, hold Shift key down and press 6.

Note that the spaces denote multiplication.

Press Enter.

Point to Create Sliders window. Create Sliders window pops up asking if you want to create sliders for a, b and p.
Click on Create Sliders. Click on Create Sliders.
Point to sliders a, p and b. Sliders are created for a, p and b.

The default setting for all three coefficients is 1.

A parabola opening upwards appears in Graphics view.

Point to vertex of parabola.

A parabola opening upwards appears in Graphics view.

a comma b correspond to the co-ordinates of the vertex.

Double click on parabola >> click on Object Properties and then on Color tab. Double click on the parabola, click on Object Properties and then on Color tab.
Select red >> close the Preferences box. Select red and close the Preferences box.
Point to the red parabola and its equation in Graphics and Algebra views. The parabola and its equation appear red in the Graphics and Algebra views.
Move boundary so you can see the equation properly. Move boundary so you can see the equation properly.
Right-click on slider a button >> check Animation On option. Right-click on slider a and check Animation On option.
Point to the parabola in Graphics view. Note the effects on the horizontal movement of the red parabola.
Right-click on slider a >> uncheck Animation On option. Right-click on slider a and uncheck Animation On option.
Right-click on slider p >> check Animation On option. Right-click on slider p and check Animation On option.
Point to parabola in Graphics view. Note the effects on the shape and orientation of the parabola.
Right-click on slider p >> uncheck Animation On option. Right-click on slider p and uncheck Animation On option.
Right-click on slider b >> check Animation On option. Right-click on slider b and check Animation On option.
Point to the parabola. Note the effects on the vertical movement of the parabola.
Right-click on slider b >> uncheck Animation On option. Right-click on slider b and uncheck Animation On option.
Point to sliders a, p and b (all = 1) and the red parabola c in Graphics view.

Click on parabola c in Graphics view and note highlighting of equation c in Algebra view.

Point to equation c: (x2-2x-4y) = -5 in Algebra view.

Note that when a, p and b are equal to 1, the red parabola c is described by equation c.

Click on parabola c in Graphics view and note highlighting of equation c in Algebra view.

Equation c is given by x squared minus 2x minus 4y equals minus 5.

Type Focus(c) in input bar>> press Enter.

Point to point A in Graphics view.

In input bar, type Focus c in parentheses.

Press Enter.

Focus is drawn at point A in Graphics view.

Point to the coordinates of point A, the Focus, in Algebra view. The coordinates of Focus of parabola c, which is point A, appear in Algebra view.
Type Vertex(c) in input bar>> press Enter.

Point to point B in Graphics view.

In input bar, type Vertex c in parentheses.

Press Enter.

Vertex is drawn at point B in Graphics view.

Point to the coordinates of point B in Algebra view. The coordinates of Vertex of parabola c, which is point B, appear in Algebra view.
Type Directrix(c) in input bar >> press Enter.


Point to Directrix in Graphics view.

In input bar, type Directrix c in parentheses.

Press Enter.

Directrix appears as a line along x axis in Graphics view.

Point to the equation, y=0, in Algebra view. The equation for the Directrix of parabola c, y equals 0, appears in Algebra view.
Double click on Directrix in Graphics view >> Redefine text box >> Object Properties >> Color tab. Double click on Directrix in Graphics view.

In the Redefine text box, click on Object Properties, then the Color tab.

In the left panel, point to highlighted Directrix, identify Focus and Vertex created for parabola c. In the left panel, note that the Directrix is highlighted.

Identify Focus and Vertex created for parabola c.

Click on each one to highlight while pressing the Control key. While pressing the Control key, click and highlight Focus and Vertex.
Click on red. Click on red.
Close the Preferences box. Close the Preferences box.
Point to Focus, Vertex and Directrix and their co-ordinates and equation in Graphics and Algebra views. For parabola c, Focus, Vertex and Directrix and their coordinates and equation appear red.
Show Geogebra window with parabolas c and d.

In Algebra view, point to equation d and in Graphics view, point to parabola d.

Follow the earlier steps to construct parabola d.
Let us summarize.
Slide Number 6

Summary

In this tutorial, we have learnt how to use GeoGebra to:
  • Study the standard equations and parts of a parabola
  • Construct parabolas
Slide Number 7

Assignment

Try these steps to construct parabolas with:

Focus (6,0) and directrix x = -6

Focus (0,-3) and directrix y = 3

Find their equations.

As an assignment:

Try these steps to construct parabolas with these foci and directrices.

Find their equations.

Slide Number 8

Assignment

Find the coordinates of the foci >> length of latus recti for these parabolas.

Also, find the equations of the axes of symmetry and directrices.

y2 = 12x

x2 = -16y

As an assignment:

Find the coordinates of the foci and length of the latus recti for these parabolas.

Also, find the equations of the axes of symmetry and directrices.

Slide Number 9

About Spoken Tutorial Project

The video at the following link summarizes the Spoken Tutorial Project.

Please download and watch it.

Slide Number 10

Spoken Tutorial workshops

The Spoken Tutorial Project team conducts workshops and gives certificates.

For more details, please write to us.

Slide Number 11

Forum for specific questions:

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

Please post your timed queries on this forum.
Slide Number 12

Acknowledgement

Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Madhurig, Vidhya

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