project work for yhong

8/9/2019 Project Work for Yhong

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PROJECT WORK FOR

ADDITIONAL MATHEMATICS2010

CURICULUM DEVELOPMENT DIVISION

MINISTRY OF EDUCATION MALAYSIA

PROJECT WORK 1

Name:

Form:

i\c number:

Subject teacher:


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INTRODUCTION

We students taking Additional Mathematics are required to carry out a project work

while we are in Form 5.This year the Curriculum Development Division, Ministry ofEducation has prepared four tasks for us.We are to choose and complete only ONE

task based on our area of interest. This project can be done in groups or individually,

and I gladly choose to do this individually. Upon completion of the Additional

Mathematics Project Work, we are to gain valuable experiences and able to:

Apply and adapt a variety of problem

Solving strategies to solve routine and non-

Routine problems

Experience classroom environments which

Are challenging, interesting and meaningfulAnd hence improve their thinking skills

Experience classroom environments where

Knowledge and skills are applied in meaningful ways in solving real-life problems.

Experience classroom environments where

Expressing ones mathematical

Thinking, reasoning and communication are

Highly encouraged and expected

Experience classroom environments that

Stimulates and enhances effective learning.

Enhance acquisition of mathematical

Knowledge and skills through problem-

solving in ways that increase interest and

confidence

Prepare ourselves for the demand of our

Future undertakings and in workplace

Realize that mathematics is an important

And powerful tool in solving real-life

Problems and hence develop positiveAttitude towards mathematics

Train ourselves not only to be independent

Learners but also to collaborate, to

cooperate, and to share knowledge in an

engaging and healthy environment

Use technology especially the ICT


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Appropriately and effectively

Train ourselves to appreciate the intrinsic

Values of mathematics and to become more

Creative and innovative

Realize the importance and the beauty ofMathematics

APPRECIATION

Alhamdullilah,thank you to Allah for giving the will to me to complete this Additional

Mathematics project. Secondly, I would like to thank the principle of Sekolah MenegahTeknik Klang, Pn. Hajah Fuzyah bt. Abdullah for giving me the permission to do my this

Additional Mathematics Project Work. I also like to thank my Additional Mathematics

teacher, Pn. Nor Azipah for the guide and giving useful and important information for me

to complete this project work. Besides that, I would like to thank my parents for their

support and encouragement. Lastly, a special thanks to all my friends for their help and

cooperation in searching for information and completing this project work.


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A BRIEF HISTORY OF QUADRATIC

FUNCTION

Origin of word

The adjective quadratic comes from the Latin word quadratum for square. A term

like x2 is called a square in algebra because it is the area of a square with side x.

In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral

and quadrant. Quadratum is the Latin word for square because a square has four

sides.

Roots

The roots (zeros) of the quadratic function

are the values ofx for whichf(x) = 0.

When the coefficientsa, b, and c, are real orcomplex, the roots are


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where the discriminant is defined as

Form of quadratic functions

quadratic function can be expressed in three formats:[1]

yis called the general form,

y is called the factored form, where x1 andx2 arethe roots of the quadratic equation, it is used in logistic map

y is called the vertex form andy (also the standard form) where h and kare the x and y coordinates of the

vertex, respectively.

To convert the general form to factored form, one needs only the quadratic

formula to determine the two roots r1 and r2. To convert the general form to

standard form, one needs a process called completing the square. To convert the

factored form (or standard form) to general form, one needs to multiply, expand

and/or distribute the factors


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Graph

Regardless of the format, the graph of a quadratic function is aparabola (as shown

above).

y If (or is a positive number), the parabola opens upward.y If (or is a negative number), the parabola opens downward.

The coefficient a controls the speed of increase (or decrease) of the quadratic

function from the vertex, bigger positive a makes the function increase faster and

the graph appear more closed.

The coefficients b and a together control the axis of symmetry of the parabola (also

thex-coordinate of the vertex) which is at x = -b/2a.

The coefficient b alone is the declivity of the parabola as it crosses the y-axis.

The coefficient c controls the height of the parabola, more specifically, it is the

point where the parabola crosses the y-axis.

xintercepts

Inspection of the factored form shows that thex-intercepts of the graph are given

by the roots of the quadratic function. These are simply the x-coordinates for

which the function equals zero.


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Vertex

The vertex of a parabola is the place where it turns, hence, it's also called the

turning point. If the quadratic function is in vertex form, the vertex is . By

the method of completing the square, one can turn the general form

into

so the vertex of the parabola in the general form is

If the quadratic function is in factored form

the average of the two roots, i.e.,

is the x-coordinate of the vertex, and hence the vertex is

The vertex is also the maximum point if or the minimum point if

The vertical line

that passes through the vertex is also the axis ofsymmetry of the parabola.


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The square root of quadratic function

The square root of a quadratic function gives rise either to an ellipse or to a

hyperbola.If then the equation describes a hyperbola.

The axis of the hyperbola is determined by the ordinate of the minimum point of


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the corresponding parabola .

If the ordinate is negative, then the hyperbola's axis is horizontal. If the ordinate is

positive, then the hyperbola's axis is vertical.

If then the equation describes either an ellipse or

nothing at all. If the ordinate of the maximum point of the corresponding parabola

is positive, then its square root describes an ellipse, but if the

ordinate is negative then it describes an empty locus of points

Iteration

Given anf(x) = ax2

+ bx + c, one cannot always deduce the analytic form off(n)

(x),

which means the nth iteration off(x). (The superscript can be extended to negative

number referring to the iteration of the inverse off(x) if the inverse exists.) But

there is one easier case, in whichf(x) = a(x x0)2

+ x0.

In such case, one has

,

where

and .

So by induction,


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can be obtained, where g(n)

(x) can be easily computed as

.

Finally, we have

,

in the case off(x) = a(x x0)2

+ x0.

See Topological conjugacy for more detail about such relationship between fand g.

And see Complex quadratic polynomial for the chaotic bahavior in the general

interation


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Biviriate(two variables)quadratic function

A bivariate quadratic function is a second-degree polynomial of the form

Such a function describes a quadratic surface. Setting equal to zero

describes the intersection of the surface with the plane , which is a locus of

points equivalent to a conic section.


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Minimum and maximum

If the function has no maximum or minimum, its graph forms anhyperbolicparaboloid.

If the function has a minimum ifA>0, and a maximum ifA0 and a maximum if

A


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Part2


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Conclusion

After doing research,answering questions,drawing graphs and some problem

solving, I saw that the usage of quadratic function is important in daily life.It is not

just widely used in making buildings but also in measurement of the surrounding

like the building in the planet.Especially making new building and tower.In

conclusion,quadratic function is a daily life nessecities.Without it,constructing work

cant be conducted,the measurement of building cant be interpret and many

more.So,we should be thankful of the people who contribute in the idea of quadratic

function

Reflection

After spending countless hours,days and night to finish this project and also

sacrificing my time video games and mangas in this mid year holiday,there are

several things that I can say...


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Additional Mathematics...

From the day I born...

From the day I was able to holding pencil...

From the day I start learning...

And...

From the day I heard your name...

I always thought that you will be my greatest obstacle and rival in excelling

in my life...

But after countless of hours...

Countless of days...

Countless of nights...

After sacrificing my precious time just for you...

Sacrificing my ComputerGames...

Sacrificing my Video Games...

Sacrificing my Facebook...Sacrificing my Internet...

Sacrifing my Anime...

Sacrificing my Manga...

I realized something really important in you...

I really love you...

You are my real friend...

You my partner...

You are my soulmate...I LOVE U ADDITIONAL MATHEMATICS...

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