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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...