Applications-of-GeoGebra/C2/Roots-of-Polynomials/English - Revision history

Madhurig at 11:33, 28 June 2018

2018-06-28T11:33:29Z

Madhurig at 11:27, 28 June 2018

2018-06-28T11:27:13Z

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Madhurig at 10:38, 28 June 2018

2018-06-28T10:38:57Z

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Vidhya at 05:37, 19 June 2018

2018-06-19T05:37:44Z

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Vidhya at 06:50, 17 May 2018

2018-05-17T06:50:43Z

Nancyvarkey at 02:32, 13 March 2018

2018-03-13T02:32:08Z

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Vidhya: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Roots of Polynomials'''. |- | | '''Slide..."

2018-03-09T10:14:52Z

Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Roots of Polynomials'''. |- | | '''Slide..."

New page

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Roots of Polynomials'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn:
To plot graphs of '''polynomial''' equations

About '''complex numbers''', '''real''' and '''imaginary roots'''

To find '''extrema''' and '''inflection points'''
|-
| | '''Slide Number 3'''
'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this tutorial, you should be familiar with '''GeoGebra''' interface

Basics of '''coordinate system'''

'''Polynomials'''

If not, for relevant tutorials, please visit our website.
|-
| | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
| |
| | Let us begin with the '''binomial theorem'''.
|-
| | '''Slide Number 5'''

'''Binomial Theorem'''

'''Binomial theorem''' states that if ''a, b'' Єℝ, '''index''' ''n'' is a '''positive integer''', ''0 ≤ r ≤n, then (a + b)n can be expanded as follows:''

''(a + b)n <nowiki>= </nowiki>nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn''

Reminder: ''nC1 = n!/[1! (n-1)!]''
| | '''''a''''' and '''''b''''' are '''real numbers''', '''index''' '''''n''''' is a positive integer.

'''''r''''' lies between 0 and '''''n'''''.

'''Binomial theorem''' states that '''''a''''' plus '''''b''''' raised to '''''n''''' can be expanded as shown.
|-
| | '''Slide Number 6'''

'''Quadratic Equations and Roots'''

A second degree polynomial, '''y =''' '''''a''''' '''x2+''' '''''b''''' '''x+''' '''''c''''' has roots

'''x=-''' '''''b''''' '''± sqrt{(''' '''''b''''' '''2-4''' '''''ac)/2a''''' '''}'''

where '''▲=''' '''''b2-4ac'''''

When ▲< 0, roots are complex

When ▲=0, roots are real and equal

When ▲>0, roots are real and unequal
| | '''Quadratic Equations and Roots'''

A '''2nd degree polynomial''', '''y equals''' '''''a''''' '''x squared plus''' '''''b''''' '''x plus''' '''''c''''' has '''roots''' given by values of '''''x'''''.

'''''x''''' is equal to '''ratio''' of minus '''''b''''' plus or minus '''squareroot''' of '''''b''''' '''squared''' minus 4 '''''a c''''' to 2 '''''a'''''.

Where '''determinant''' '''Delta''' is equal to '''''b''''' '''squared''' minus 4 '''''a c'''''

When '''Delta''' is less than 0, '''roots''' are '''complex'''

When '''Delta''' is equal to 0, '''roots''' are '''real''' and equal

When '''Delta''' is greater than 0, '''roots''' are '''real''' and unequal
|-
| | '''Slide Number 7'''

'''Quadratic Equations and Roots'''

When roots are real, '''''ax2+b''''' '''x+''' '''''c''''' '''=0''' has extremum '''(xv, yv)'''

'''xv = -''' '''''b/2a''''' and '''yv=''' '''''a''''' '''xv2+''' '''''b''''' '''xv+''' '''''c'''''
| | '''Quadratic Equations and Roots'''

When '''roots''' are '''real''', '''''ax''''' '''squared plus''' '''''b x''''' '''plus''' '''''c''''' equals 0 has '''extremum''' '''''xv''''' '''comma''' '''''yv'''''

'''''xv''''' equals '''minus''' '''''b''''' '''divided by 2''' '''''a''''' and '''''yv''''' '''equals''' '''''axv''''' '''squared plus''' '''''bxv''''' '''plus''' '''''c'''''
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.
|-
| | Click on '''View''' >> select '''CAS'''.

| | Click on '''View''' tool and select '''CAS''' to open the '''CAS''' view.
|-
| | In line 1 in '''CAS view''', type '''f(x):=x^2-2x-3''' >> press '''Enter'''.
| | In line 1 in '''CAS''' view, type the following line.

'''f x''' in parentheses '''colon equals x caret 2 minus 2 space x minus 3'''.

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

The space indicates multiplication.

Press '''Enter'''.
|-
| | Drag boundary to see '''Algebra''' view properly.
| | Drag boundary to see '''Algebra''' view properly.
|-
| | Point to the '''equation f(x)''' appearing in '''Algebra''' view.

Point to '''exponent''' 2 in '''f(x)'''.
| | Observe the '''equation f of x''' in '''Algebra''' view.

The '''degree''' of this '''quadratic polynomial f of x''' is 2.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag boundary to see '''Graphics''' view properly.
|-
| | Click in '''Graphics''' view to see '''Graphics View''' toolbar.
| | Click in '''Graphics''' view to see '''Graphics View''' toolbar.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view to see the minimum '''vertex''' of '''parabola f'''.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view to see the minimum '''vertex''' of '''parabola f'''.
|-
| | Click on '''Move Graphics View''' tool and click in '''Graphics''' background.

When hand symbol appears, drag '''Graphics''' view so you can see parabola '''f'''.
| | Click on '''Move Graphics View''' tool and click in '''Graphics''' background.

When hand symbol appears, drag '''Graphics''' view so you can see parabola '''f'''.
|-
| | Drag boundaries to see '''CAS''' view properly.
| | Drag boundaries to see '''CAS''' view properly.
|-
| | Type '''Root(f)''' in line 2 of '''CAS''' view >> press '''Enter'''.
| | In line 2 of '''CAS''' view, type '''Root f''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''roots''' in '''CAS''' view.
| | The '''roots''' appear below, in the same box, in curly brackets.
|-
| | Point to '''roots''' in '''Graphics''' view.
| | Note that these are the '''x-intercepts''' of parabola '''f''' in '''Graphics''' view.
|-
| | Type '''Extremum(f)''' in line 3 of '''CAS''' view >> press '''Enter'''.
| | In line 3 of '''CAS''' view, type '''Extremum f''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''extremum''' in '''CAS''' view.
| | The '''extremum''' appears below, in the same box, in curly brackets.
|-
| | Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
| | Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
|-
| | In line 4 in '''CAS''' view, type '''g(x):=x^2+5x+10''' >> press '''Enter'''.
| | In line 4 in '''CAS''' view, type the following line.

'''g x''' in parentheses '''colon equals x caret 2 plus 5 space x plus 10'''.

Press '''Enter'''.
|-
| | Drag boundary to see '''Algebra''' view properly.

Point to the '''equation g(x)''' appearing in '''Algebra''' view.
| | Drag boundary to see '''Algebra''' view properly.

Observe the '''equation g of x''' in '''Algebra''' view.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag boundary to see '''Graphics''' view properly.
|-
| | Uncheck '''f of x''' in '''CAS''' view.

Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.
| | Uncheck '''f of x''' in '''CAS''' view.

Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.
|-
| | Click and drag '''Graphics''' view so you can see parabola '''g'''.
| | Click in and drag '''Graphics''' view so you can see parabola '''g'''.
|-
| | Again, drag boundary to see '''CAS''' view properly.
| | Again, drag boundary to see '''CAS''' view properly.
|-
| | Type '''Root(g)''' in line 5 of '''CAS''' view >> press '''Enter'''.
| | In line 5 of '''CAS''' view, type '''Root g''' in parentheses.

Press '''Enter'''.
|-
| | Point to empty curly brackets for '''roots''' in '''CAS''' view.
| | Empty curly brackets appear below.

Parabola '''g''' does not have any '''real roots''' as it does not intersect '''x axis''' at all.

'''Roots''' are said to be '''complex'''.
|-
| | Type '''Extremum(g)''' in line 6 of '''CAS '''view >> press '''Enter'''.
| | In line 6 of '''CAS''' view, type '''Extremum g''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''extremum''' in '''CAS''' view.
| | The '''extremum''' appears below, in the same box, in curly brackets.
|-
| | Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
| | Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
|-
| | Point to '''Evaluate''' tool.

Point to '''extremum''' in form of '''fractions'''.
| | While '''Evaluate''' tool is highlighted in '''CAS View''' toolbar, the '''extremum''' appears as '''fractions'''.

'''Minus five divided by 2 comma 15 divided by 4'''.
|-
| | Click on the '''extremum''' in line 6 and click on '''Numeric''' tool.

Point to '''extremum''' in form of '''decimals'''.
| | In line 6, click on the '''extremum''' and click on '''Numeric''' tool.

The '''extremum '''now appears in '''decimal''' form.

'''Minus 2 point 5 comma 3 point 7 5'''.
|-
| |
| | Let us look at '''complex numbers'''.
|-
| | '''Slide Number 8'''

'''Complex numbers, XY plane'''

As we know,
A '''complex number''' is expressed as '''''z = a + ib''''': where ''''''a'''''' is the real part, ‘'''''bi’''''' is '''imaginary '''part, and '''a''' and '''b''' are constants.

'''Imaginary number, ''i'' '''= sqrt{-1}

In the '''XY plane''', '''''a + ib''''' corresponds to the point ('''a, b''').

In the '''complex plane''', '''x axis''' is called '''real axis''', '''y axis''' is called '''imaginary axis'''.
| | '''Complex numbers, XY plane'''

As we know,
A '''complex number''' is expressed as '''''z''''' '''equals''' '''''a''''' '''plus''' '''''ib'''''.

'''''a''''' is the '''real''' part; '''''bi''''' is '''imaginary '''part;'''''a''''' and '''''b''''' are '''constants'''

'''''i''''' is '''imaginary number''' and is equal to '''square root of minus 1'''.

In the '''XY plane''', '''''a''''' '''plus''' '''''ib''''' corresponds to the point '''''a''''' '''comma''' '''''b'''''.

In the '''complex plane''', '''x axis''' is called '''real axis, y axis''' is called '''imaginary axis'''.
|-
| | '''Slide Number 9'''

'''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector''' with '''real axis coordinate''' '''''a''''' and '''imaginary axis coordinate''' '''''b'''''

Length of the '''vector''' '''''z''''' = |'''''z'''''| = '''''r'''''

'''''r''''' '''= sqrt (a2+b2) (Pythagoras’ theorem)'''
| | '''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector'''.

Its '''real axis coordinate''' is '''''a''''' and '''imaginary axis coordinate''' is '''''b'''''.

The length of the '''vector''' '''''z''''' is equal to the '''absolute value''' of '''''z''''' and to '''''r'''''.

According to '''Pythagoras’ theorem''', '''''r''''' is equal to '''squareroot of''' '''''a''''' '''squared plus''' '''''b''''' '''squared'''.
|-
| | '''Slide Number 10'''

'''Complex numbers, complex plane'''

'''Argument '''''ϴ''''' '''= angle between '''real axis''' and '''line segment''' connecting '''''z''''' to O '''(0,0)''' in counter-clockwise direction

'''Polar form''' of '''''z = a + ib''''' is

'''''z = r (cosϴ + i sinϴ)'''''

where '''''a= r cosϴ, b=r sinϴ'''''
| | '''Argument ''theta''''' is angle between '''real axis''' and line segment connecting '''''z''''' to '''origin'''.

It is in counter-clockwise direction.

'''Polar form''' of '''''z''''' equals '''''a''''' '''plus''' '''''ib''''' is

'''''z''''' equals '''''r times cos theta plus i sin theta'''''

where '''''a''''' is equal to '''''r cos theta''''' and '''''b is r sin theta'''''
|-
| | Show the '''GeoGebra''' window.
| | Let us go back to the '''GeoGebra interface''' we were working on.

We will now use the '''input bar''' instead of '''CAS''' view.
|-
| | Click and close '''CAS''' view.
| | Click and close '''CAS''' view.
|-
| | In '''Algebra''' view, uncheck '''g of x''' to hide it.
| | In '''Algebra''' view, uncheck '''g of x''' to hide it.
|-
| | In '''input bar''', type the following line.

'''h(x):=x^3-4x^2+x+6''' >> press '''Enter'''.

| | In '''input bar''', type the following line.

'''h x''' in parentheses '''colon equals x caret 3 minus 4 space x caret 2 plus x plus 6'''.

Press '''Enter'''.
|-
| | Drag boundaries to see '''Algebra''' and '''Graphics''' view properly.

Point to the '''equation h(x)''' appearing in '''Algebra''' view.
| | Drag boundaries to see '''Algebra''' and '''Graphics''' view properly.

Observe equation '''h of x''' in '''Algebra''' view.

Function '''h of x''' is graphed in '''Graphics''' view.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' and move background to see the graph.
| | Click on '''Move Graphics View''' and move background to see the graph.
|-
| | In '''input bar''', type '''Root(h)''' and press '''Enter'''.
| | In '''input bar''', type '''Root h''' in parentheses and press '''Enter'''.
|-
| | Point to '''co-ordinates''' of three '''roots''' ('''A, B''' and '''C''') in '''Algebra''' view.
| | The '''co-ordinates''' of three '''roots''' ('''A, B''' and '''C''') appear in '''Algebra''' view.
|-
| | Point to three '''roots''' mapped on the '''curve h of x''' in '''Graphics''' view.
| | The three '''roots''' are also mapped as '''x intercepts''' of the '''curve h of x''' in '''Graphics''' view.
|-
| | In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
| | In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
|-
| | Point to '''co-ordinates''' of two '''extrema''' ('''D'''and '''E''') in '''Algebra''' view.
| | '''Co-ordinates''' of two '''extrema''' ('''D''' and '''E''') appear in '''Algebra''' view.
|-
| | Point to two '''extrema''' mapped on the '''curve h of x''' in '''Graphics''' view.
| | The two '''extrema''' are also mapped on '''curve h of x''' in '''Graphics''' view.
|-
| | '''Slide Number 11'''

'''Point of inflection'''

'''Point of inflection''' ('''PoI''') on a curve is the point where '''curve''' changes direction.

To find co-ordinates of PoI (x,y)

Equate 2nd derivative of given function to 0

Solve to get x (x co-ordinate of PoI)

Substitute this x in original function to get y co-ordinate
| | A '''point of inflection PoI''' on a curve is the point where the '''curve''' changes its direction.

To find the '''co-ordinates''' of '''PoI''' '''''x''''' comma '''''y''''':

We equate the second '''derivative''' of the given '''function''' to 0.

Solution of this equation gives us '''x''' ('''x co-ordinate''' of '''PoI''').

Substitute this '''x''' in original '''function''' to get '''y co-ordinate'''.
|-
| | Let us find the '''point of inflection''' on '''h(x)'''.
| | Let us find the '''point of inflection''' on '''h of x'''.
|-
| | In '''input bar''', type '''Inf''' >> choose '''InflectionPoint ( <Polynomial> )''' option from '''menu'''.
| | In '''input bar''', type '''Inf''' and scroll down menu to choose '''InflectionPoint Polynomial''' option.
|-
| | Instead of highlighted '''Polynomial''', type '''h''' >> Press '''Enter'''.
| | Instead of highlighted '''Polynomial''', type '''h''' and press '''Enter'''.
|-
| | Point to the '''point of inflection''' in '''Algebra''' view.
| | In '''Algebra''' view, '''point of inflection''' appears as point '''F''', below the two '''extrema'''.
|-
| | Point to '''F''' on '''h(x)''' in '''Graphics''' view.
| | '''F''' is mapped on '''h of x''' in '''Graphics''' view.
|-
| |
| | Let us open a new '''GeoGebra''' window to use '''CAS''' for a '''cubic polynomial'''.
|-
| | Click on '''View''' tool and click on '''CAS''' to show it.
| | Click on '''View''' tool and click on '''CAS''' to show it.
|-
| | In line 1 of '''CAS''' view, type the following line.

'''i(x):=x^3-6 x^2+4 x+1''' >> press '''Enter'''.
| | In line 1 of '''CAS'' 'view, type the following line.

'''i x''' in parentheses '''colon equals x caret 3 minus 6 space x caret 2 plus 4 space x plus 1'''.

Press '''Enter'''.
|-
| | Drag boundary to see '''CAS''' view properly.
| | Drag boundary to see '''CAS''' view properly.
|-
| | Drag boundary to see '''Algebra''' view properly.
| | Drag boundary to see '''Algebra''' view properly.
|-
| | In line 2 of''' CAS''' view, type '''Root(i)''' >> press '''Enter'''.
| | In line 2 of''' CAS''' view, type '''Root i''' in parentheses and press '''Enter'''.
|-
| | Point to the three '''roots''' in '''CAS''' view.

Scroll to see them.

Point to '''Evaluate''' tool.
| | The three '''roots''' are shown below with '''square root notations'''.

Scroll to see them.

Note that the '''Evaluate''' tool is highlighted.
|-
| | In line 2, click on the '''roots''' and click on '''Numeric''' tool.

Point to the three '''roots''' in decimal form.
| | In line 2, click on the '''roots''' and click on '''Numeric''' tool.

The roots are now shown in '''decimal''' form in the next line.
|-
| | In line 4 of '''CAS''' view, type '''Extremum(i)''' >> press '''Enter'''.
| | In line 4 of '''CAS''' view, type '''Extremum i''' in parentheses and press '''Enter'''.
|-
| | Point to the two '''extrema''' in '''CAS''' view.

Scroll to see them.

Point to '''Numeric''' tool and to '''extrema''' in '''decimal''' form.
| | The two '''extrema''' points are shown below.

Scroll to see them.

As the '''Numeric''' tool was clicked, the points appear in '''decimal''' form.
|-
| | Click in and drag '''Graphics''' view so you can see '''i(x)'''.
| | Click in and drag '''Graphics''' view so you can see '''i of x'''.
|-
| | In line 5, type '''Inf''' >> choose '''InflectionPoint ( <Polynomial> )''' option from '''menu'''.
| | In line 5, type '''Inf''' and scroll down menu to choose '''InflectionPoint Polynomial''' option.
|-
| | Instead of highlighted '''Polynomial''', type '''i''' >> Press '''Enter'''.
| | Instead of highlighted '''Polynomial''', type '''i''' and press '''Enter'''.
|-
| | Point to the '''point of inflection''' in '''Algebra''' view.
| | '''Co-ordinates''' of '''point of inflection''' appear in curly brackets.
|-
| |
| | Correlate the '''degree''' of the '''polynomials''' and the number of '''roots''' seen so far.
|-
| | Point to '''CAS''', then '''Algebra''' and '''Graphics''' views.
| | Observe that '''functions''' entered in '''CAS''' appear in '''Algebra''' and '''Graphics''' views.
|-
| | Point to '''Algebra''' and '''Graphics''' views, then '''CAS''' view.
| | '''Functions''' entered in '''input bar''' appear in '''Algebra''' and '''Graphics''' views but not in '''CAS''' view.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 12'''

'''Summary'''
| | In this tutorial, we have learnt how to:

Plot graphs of '''polynomial functions''' using '''CAS''' view and '''input bar'''.

Find '''real roots, extrema''' and '''inflection point(s)'''.

'''Complex roots''' will be covered in another tutorial.
|-
| | '''Slide Number 13'''
'''Assignment'''
| | Assignment:

Plot '''graphs''' and find '''roots''', '''extrema''' and '''inflection points''' for the following '''polynomials'''.

d(x)=x2-6x+5

e(x)=3x3-2x2+0.2x-1

f(x)=-2x4-x3+3x2

g(x)=x5-7x4+9x3+23x2-50x+24

h(x)=(4x+3)/(x-1)

i(x)=(3x2-2x-1)/(2x2+3x-2)
|-
| | '''Slide Number 14'''
'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 15'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 16'''
'''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 17'''
'''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.
|-
|}

@@ Line 356: / Line 356: @@
 || The two '''extrema''' are also mapped on '''curve h of x''' in '''Graphics''' view.
 |-
-|| '''Slide Number 11'''
+|| '''Slide Number 10'''
 '''Point of inflection'''
@@ Line 408: / Line 408: @@
 || Let us summarize.
 |-
-|| '''Slide Number 12'''
+|| '''Slide Number 11'''
 '''Summary'''
@@ Line 419: / Line 419: @@
 * '''Complex roots''' will be covered in another tutorial
 |-
-|| '''Slide Number 13'''
+|| '''Slide Number 12'''
 '''Assignment'''
@@ Line 438: / Line 438: @@
 Plot '''graphs''' and find '''roots''', '''extrema''' and '''inflection points''' for the following '''polynomials'''.
 |-
-|| '''Slide Number 14'''
+|| '''Slide Number 13'''
 '''About Spoken Tutorial project'''
@@ Line 445: / Line 445: @@
 Please download and watch it.
 |-
-|| '''Slide Number 15'''
+|| '''Slide Number 14'''
 '''Spoken Tutorial workshops'''
@@ Line 452: / Line 452: @@
 For more details, please write to us.
 |-
-|| '''Slide Number 16'''
+|| '''Slide Number 15'''
 '''Forum for specific questions:'''
@@ Line 466: / Line 466: @@
 || Please post your timed queries on this forum.
 |-
-|| '''Slide Number 17'''
+|| '''Slide Number 16'''
 '''Acknowledgement'''

@@ Line 251: / Line 251: @@
-A '''complex number''' is expressed as '''''z = a + ib''''': where ''''''a'''''' is the real part, ‘'''''bi’''''' is '''imaginary '''part, and '''a''' and '''b''' are constants.
+A '''complex number''' is expressed as '''''z = a + bi''''': where ''''''a'''''' = real part, ‘'''''bi’''''' = '''imaginary '''part, and '''a''' and '''b''' are constants.
 '''Imaginary number, ''i'' '''= sqrt{-1}
-In the '''XY plane''', '''''a + ib''''' corresponds to the point ('''a, b''').
+In the '''XY plane''', '''''a + bi''''' is point ('''a, b''').
-In the '''complex plane''', '''x axis''' is called '''real axis''', '''y axis''' is called '''imaginary axis'''.
+In the '''complex plane''', '''x axis''' = '''real axis''', '''y axis''' = '''imaginary axis'''.
 |  | '''Complex numbers, XY plane'''
-A '''complex number''' is expressed as '''''z''''' '''equals''' '''''a''''' '''plus''' '''''ib'''''.
+A '''complex number''' is expressed as '''''z''''' '''equals''' '''''a''''' '''plus''' '''''bi'''''.
@@ Line 271: / Line 271: @@
-In the '''XY plane''', '''''a''''' '''plus''' '''''ib''''' corresponds to the point '''''a''''' '''comma''' '''''b'''''.
+In the '''XY plane''', '''''a''''' '''plus''' '''''bi''''' corresponds to the point '''''a''''' '''comma''' '''''b'''''.
@@ Line 304: / Line 304: @@
 '''Complex numbers, complex plane'''
-'''Argument '''''ϴ''''' '''= angle between '''real axis''' and '''line segment''' connecting '''''z''''' to O '''(0,0)''' in counter-clockwise direction
+'''Argument '''''ϴ''''' '''= angle between '''real axis''' and '''line segment''' connecting '''''z''''' to O '''(0,0)'''; CCW
-'''Polar form''' of '''''z = a + ib''''' is
+'''Polar form''' of '''''z = a + bi''''' is
 '''''z = r (cosϴ + i sinϴ)'''''
@@ Line 316: / Line 316: @@
-'''Polar form''' of '''''z''''' equals '''''a''''' '''plus''' '''''ib''''' is
+'''Polar form''' of '''''z''''' equals '''''a''''' '''plus''' '''''bi''''' is
 '''''z''''' equals '''''r times cos theta plus i sin theta'''''
-where '''''a''''' is equal to '''''r cos theta''''' and '''''b is r sin theta'''''
+Where '''''a''''' is equal to '''''r cos theta''''' and '''''b is r sin theta'''''
 |-
 |  | Show the '''GeoGebra''' window.
@@ Line 347: / Line 347: @@
 Point to the '''equation h(x)''' appearing in '''Algebra''' view.
-|  | Drag boundaries to see '''Algebra''' and '''Graphics''' view properly.
+|  | Drag boundaries to see '''Algebra''' and '''Graphics''' views properly.
 Observe equation '''h of x''' in '''Algebra''' view.
@@ Line 360: / Line 360: @@
 Click in '''Graphics''' view.
 |-
-|  | Click on '''Move Graphics View''' and move background to see the graph.
+|  | Click on '''Move Graphics View''' and move '''Graphics''' background to see the graph.
 |  | Click on '''Move Graphics View''' and move '''Graphics''' background to see the graph.
 |-
@@ Line 394: / Line 394: @@
 Substitute this x in original function to get y co-ordinate
-|  | A '''point of inflection PoI''' on a curve is the point where the '''curve''' changes its direction.
+|  | '''Point of inflection'''
-−
 A '''point of inflection PoI''' on a curve is the point where the '''curve''' changes its direction.
-To find the '''co-ordinates''' of '''PoI''' '''''x''''' comma '''''y''''':
-We equate the second '''derivative''' of the given '''function''' to 0.
+To find the '''co-ordinates''' of '''PoI''' '''''x''''' comma '''''y''''', we equate second '''derivative''' of the given '''function''' to 0.
@@ Line 506: / Line 505: @@
 '''Summary'''
-|  | In this tutorial, we have learnt how to:
+|  | In this tutorial, we have learnt to:
-*Plot graphs of '''polynomial functions''' using '''CAS''' view and '''input bar'''.
+*Plot graphs of '''polynomial functions''' using '''CAS''' view and '''input bar'''
-*Find '''real roots, extrema''' and '''inflection point(s)'''.
+*Find '''real roots, extrema''' and '''inflection point(s)'''
-*'''Complex roots''' will be covered in another tutorial.
+*'''Complex roots''' will be covered in another tutorial
 |-
 |  | '''Slide Number 13'''