EUROPEAN SCHOOL Mathematics Higher Level Portfolio Type 1 SHADOW FUNCTIONS Candidate Name: Emil Abrahamyan candidate Number: 006343-021 Supervisor: Avtandil Gagnidze Session Year: 2013 May Type 1: Shadow Functions The Aim of the Investigation: The overall aim of this investigation is to investigate different polynomials with different powers and create shadow function for each one.

Afterwards identify the real and imaginary components of complex zeros from the key points along the x-axis using the method of shadow functions and their generators.Technology Used: Technology that had been used is shown below Autograph (Version 3. 3) Graphing Display Calculator Tl-84 Plus Texas Instruments 2) Defining terms:’ Quadratic, cubic, quartic functions are members of the family of polynomials. A quadratic function is a function of the form constants and A cubic function is a function of the form A quartic function is a function of the form where are constants and Complex numbers is any number of the form where , are are real and The vertex of parabola is point where the parabola crosses its axes of symmetry.

d Urban, P. , Martin, D. , Haese, R. , Haese, S. , Haese M.

and Humphries, M. (2008) Mathematics HL (Core). 2 ed. ; Adelaide Airport: Haese & Harris Publications 2 Processing: , where is the transformation of the graph by a vector as shown in the Diagram 1. As the coordinates of the vertex of will be Diagram 1 clearly shows that then the coordinates of the vertex of doesn’t have any real solutions, as it doesn’t intersect x-axis. In order to find the imaginary solutions of , the following equation should be solved. where a,b 3 The shadow function to in Diagram 2. s another quadratic which shares the same vertex as as shown Diagram 2 The properties of Function re illustrated in Table 1.

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Equation Coordinates of Vertex Zeros Table 1 4 With the purpose of finding any patterns between , and , various values of be used in order to generate pairs of , and as shown below. Values of and will Diagram 3 Comments As seen from the graph has downwards concavity Diagram 4 illustrates that cuts the x-axis at and cuts the x-axis at , which , which means that it has zeroes means that it has zeros and .As (upwards concavity) doesn’t cross the x-axis, it Again doesn’t cross x-axis and it has hasn’t any real zeros. It has imaginary zeroes imaginary zeroes The equation of shadow generating function is , which means that the position of shadow generating function depends on the positions of Table 2 5 Now other values of will be tested as shown in Table 3 -2 -5 6 8 Diagram 5 Diagram 6 has downwards Diagram 6 illustrates that concavity and cuts the x-axis at , which means that it has , which means that it has zeros and zeroes and .Again doesn’t cross x-axis and it has imaginary As , which has upwards concavity, doesn’t cross zeros the x-axis it hasn’t any real zeros. It has imaginary zeroes The equation of shadow generating function is Table 3 Observations and Conclusions: As seen from the graphs the shadow generating function , which is parallel to x-axis, passes or . This means that is the y-coordinate of the vertex. Moreover as to any value there are corresponding values of symmetrical to , therefore the central point of function .