- Level: International Baccalaureate
- Subject: Maths
- Word count: 881
Quadratic Polynomials. Real and Imaginary components
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Introduction
MATHS IA Alex Chen
PART A (Quadratic Polynomials)
The investigation is to find out if the zeros and to determine the real and imaginary components of the complex zeros of
.
From the function given,
The coordinates of the vertex is
by using the Quadratic equation:
where
Hence,
has zeros
, and
By subbing in different numbers of
into the equation:
For:
, it is given that
, which is equal to
:
:
Value of a | value of b | value of y1 |
1 | 1 | |
2 | 2 | |
3 | 3 | |
4 | 4 |
Value of a | value of b | value of y1 |
1 | 1 | |
2 | 2 | |
3 | 3 | |
4 | 4 |
For:
:
:
Value of a | value of b | value of y1 |
1 | 1 | |
2 | 2 | |
3 | 3 | |
4 | 4 |
Value of a | value of b | value of y1 |
1 | 1 | 2 |
2 | 2 | |
3 | 3 | |
4 | 4 |
Middle
We know that
has zeros
, while
has opposite concavity to
,which is in the form
.
From the graph, it can be seen that,
is a reflection of
,
Therefore, the equation of the quadratic
is :
.
When
As the graph shown above,
and
have point of intersection, The shadow generating function is
.
Therefore, the shadow generating function for this quadratic is
To express
in terms of
,
,
.
By substitution into this function:
As
Coordinates of the vertex,
Shadow function:
Shadow root:
Complex zeros:
The reflection
As the above results shown, in the complex zeros:
the imaginary components is
PART B (Cubic Polynomials)
The investigation
Conclusion
.
It can be written in the below format:
For the generating function of
Firstly, it’s slope needs to be found and the intersection point of
,
By using the equation:
And the generating formula is calculated as:
As the above graph shown, the cubic function
and
,
cuts through between the intersection points of
and
,
fits the result.
In general, the cubic function
is a reflection of
.
Considering the derived function,
,
the function
Again, considering the derived function,
,
As the above diagram shows,
from point E to D and point F to D is the distance of the quadratic function
,
therefore the derived function
works in a quadratic function.
As the above diagram shows,
from point a to b and point c to d is the distance of the
,
therefore the derived function
works in cubic function.
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