Logan’s Logo

Abstract

This internal assessment focuses on functions and areas under curves. The task at hand was to develop models of functions to best fit the characteristics/behaviours of the curve of Logan’s logo. Using an appropriate set of axes data points of the curve were measured using a ruler and then identified and recorded. 11 data points were recorded for each curve. The main objective of this IA was to determine which model, cubic or sin, would be the best fit. It was observed that the cubic function was superior. Another task was to refine the model function to fit on a t-shirt and a business card. The area between the two curves had to be calculated. In doing this assignment many pieces of technology were incorporated when doing this assignment, ranging from a pencil and a ruler to calculators and even spreadsheets and other programs.

Introduction:

A diagram of a 10cm by 10cm square is divided into three regions by two curves. The logo is the shaded region between the two curves. This investigation is aimed at answering several questions but mainly to develop mathematical functions to model the two curves represented by the two curves. Some key terms that should be understood are: sin function, cubic function, MAE (Mean Absolute Error). These terms will be elaborated on later on this assignment.

A few data points were taken from the curves, with the base of the square representing the x-axis and each unit being one centimetre, and the y-axis represented as the left side of the square with each unit being a centimetre as well. The x-axis begins at 0 and goes on till 10 centimetres; the same is applicable for the y-axis. The lower curve is denoted at f(x) and the upper curve denoted as g(x). The following points were observed using a ruler

Table 1: observed data points for the curve f(x)

Table 2: observed data points for the curve g(x)

The points of these curves are graphed on the following graph:

Graph 1: Observed points of logo

In the graph above the points were plotted in a coordinate system with a domain of and a range of, this was done according to the size of the diagram of the logo given. The points were plotted on a graph using the program Graphamatica. These points were then connected using a best fit curve with the program Logger Pro. By graphing the best fit curve line we can see what type of function this curve might be. This graph can be seen below

Graph 2: Points of Graph 1 connected

It should be noted that the curve is not very smooth but rather rigid this is because these are not the exact points of the logo these are the points observed using a ruler, making the values slightly inaccurate resulting in a very rigid curve rather than a smooth one. Nonetheless an idea of the characteristics of the curve can be formulated. The curves best fit the family of periodical functions such as sine and cosine, as well as a polynomial function with an order of 3 or greater such

For the purpose of simplicity only a sine and cubic function model will be used to see which is appropriate to represent the behaviour of the two curves of the logo.

Cubic Model:

A cubic function has the generic function of: . Where and are the variables representing the distance horizontally and vertically from the origin, respectively. The parameters and transform the graphs in to best fit the observed points. It should be noted that the cubic model has the constraints because there is no information regarding the curves exceeding the set boundaries. In order to find a function that showcases the behaviours of the curves, the values of the parameters a, b, c, d have to be calculated. This can be calculated by solving a simultaneous equation for four equations, since there are four unknowns. It is best if the points chosen are further apart from each other good idea to select points which are further away from each other. This comparison is made below:

Lower Curve

From here we can use matrices to solve for a, b, c, and d

This gives the function: f(x)= 0.0267x3+0.0520x2-0.0232x+0.3544

However if the points chosen are more wide-spread different values for the parameters are obtained:

This gives the function: f(x)= -0.0375x3+0.4617x2-0.6900x+0.5275

However if more points are incorporated, a better fit line is obtained. This cannot be attained by using matrices because in order to obtain the inverse of a matrix it must be a square. In order to calculate this we must use a different method, a method that incorporates all these points, we will now use the cubic regression tool on a Ti-84 Plus Graphical Display Calculator (GDC). To calculate this we must upload the 11 points observed, this is done by first pressing the “STAT” button followed by “1:Edit”. A table is then brought to the screen, this will be filled with the x and y values observed. The L1 table corresponds to the x values and the L2 for the y values. Once completed the table is as follows: