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. 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 . nd these two points are belongs to shadow generating The observations made above and the fact of symmetry of functions with respect to shadow generating function, enabled to come up with a general statement shown below Proof: As shown above the shadow generating function is equal to As the function therefore and its shadow function are symmetrical in respect of shadow function , hence 7 The next part of the investigation is to illustrate how the zeroes of may be helpful in determination of the real and imaginary components of the complex zeroes of .
One of the forms of representation of quadratic function is a completing square form as shown below where a , b Consequently the shadow function of will have the form As the shadow function cuts the x-axis, its roots can be found In order to carry this investigation, the real and complex roots of function and its hadow function should be represented on the same graph, in order to make findings more obvious. In order to illustrate the complex zeros of , the function of and its zeroes should be rotated by 90 degrees among the x-axis.
Afterwards, as in Argand Diagram, the value of y-axis will become the imaginary part and the value of x-axis will become the real part of the complex root as shown in Diagram 7. The two zeroes of A(a-b) and B (a+b) are rotated 90 degrees from the centre E (a,O). The result is two points C and D with coordinates (a,b) and (a,-b) respectively. As the roots are rotated the plane converted to complex plane (also known as Argand Diagram), x therefore the value of y-axis became the imaginary part and the value of x-axis became the real part.
The vectors show two complex numbers, therefore the complex zeroes of Diagram 7 By investigating further, the method of shadow functions and their generators can be applied to cubics. The shadow function in this case is another cubic function which shares two points with Table 4 The shadow generating function in this case passes through the two points of intersection as shown in Diagram 8.