International Baccalaureate: Maths
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Ib math HL portfolio parabola investigation
In the 1st part of my portfolio I will only consider a>o, giving me a minimum point for each parabola rather than a maximum. The 2nd would include values of a < 0. We know that for a graph to have its turning point in the first quadrant and a>0, it should have no real roots. This means that the parabola would not cut with the xaxis and its determinant would be greater than zero. Hence: ,where a >0 .
 Word count: 3129

Investigating Sin Functions
Y = 2sin(x) Y = sin (x) Now, as we can see from comparing these graphs, y= 2sin(x) is a vertically stretched version of the original basic function y=sin (x), because the A (amplitude) was higher. This contention is supported even further as we continue to examine graphs with higher amplitudes. If we graph again with FooPlot, we get the following: Y= 5 sin(x) Y = 2 sin(x) As we compare the new graph with a higher Amplitude (y=5 sin(x)), we can see a tremendous, more pronounced difference from the previous graph than we did above because this time, there is a greater increase in the amplitude "A".
 Word count: 2369

Population trends in China
� y = p (1+r) n � y = ae (kt) These can be used as they include in the equation population factors, where the year can be included in them as well as interpreting the graph with them. By using the model of y = ar (n1) which could be used in other terms as: y = ar t (t for time) I may develop this model with my data. Again, I am referring to the years in terms of being from year 0 to 45 instead of from 1950 to 1995.
 Word count: 2103

Artificial Intelligence & Math
claim that exploitations of the Internet have doubled from 1999. If such a trend continues the Internet will be too unsafe to be used without taking an unreasonable risk. As the Internet becomes more commonly used as a tool in criminal, terrorist and cyber terrorist activity governments around the world are beginning to reevaluate their stance on its surveillance (Miller 2001,Kane 2002). Some trends but no developments. The ISP, provider of direct access to the Internet backbone, is essential in the surveillance. All Australian use of the Internet will be continually monitored by automated ISP computer mainframes for specific security flags.
 Word count: 5042

Logarithm Bases Math IA
2) Cleary, the pattern is noticeable as the sequence goes on. Therefore, I was able to find the term for each sequence, writing it in the form where . For the general logarithmic sequence, the term is the following. ..., That being the case, the nth term for the three examples of logarithmic sequences are the following: 1) 2) 3) To justify my answer using technology, I used excel to verify that the pattern continues (see appendix 1). By carrying out the pattern to where n=50, I was able to confirm that the pattern continues.
 Word count: 828

Portfolio: Body Mass Index
To be more sure, I drew a simple Gaussian function over the graph, which I already have had. The new graph looks as follows (the red line is the function that the given data form and the black line is the Gaussian function): The exact equation of the Gaussian function, which fits the data graph, is: y =, where A=6,933 B= 5,510 C=8.846 D=22.15 It must be noted that it is highly possible the Gaussian function to be inappropriate if the data is for elder people.
 Word count: 733

Infinite Summation
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 Word count: 1704

FISHING RODS
Also the placement of the guides (distance) from the tip does not follow a regular pattern. This might make it difficult to achieve a function that satisfies all the points. Graph 1. Plotted graph of number of guides and their respective distances from the tip on Leo's fishing rod it can also be seen from Graph 1 that the plotted data of Leo's fishing rod begins from (1, 10) and goes up to (8,149) forming an ascending pattern. Thus this pattern indicates that the constant of the highest index on each of the model functions must be positive.
 Word count: 1974

Stellar numbers
+ c 3 = 4a + 2b + c Tn = 6 n = 3 6 = a(3)2 + b(3) + c 6 = 9a + 3b + c After finding these three different equations, I solved to find the values of a, b and c. 1 = a + b + c 3 = 4a + 2b + c  3 = 4a + 2b + c  6 = 9a +3b + c ______________ _______________ 2 = 3a + b 3 = 5a + b 2 = 3a + b  3 = 5a + b ______________ 1 = 2a a = 0.5 2 = 3a + b 2 = 3(1/2)
 Word count: 1841

Maths Investigation: Pascals Triangles
You can see the pattern 2,3,4. I can predict that the next one will be five and the one after six and so on. Each number in the triangle is the sum of the two directly above it. So the next row will be. 1, 5, 10, 10, 5, 1. this is because you put 1 first then add 1 + 4 which is 5, then you add 4 + 6 which is 10 and 4 + 6 again. then you 1+4 again and you add a 1 to the outside.
 Word count: 1681

Stopping Distances
The fact that a linear function resembles this situation in a very accurate manner means that the speed to thinking distance ratio is constant. Speed (x) Vs. Braking Distance (y) Points plotted on the graph: (32, 6) (48, 14) (64, 24) (80, 38) (96, 55) (112, 75) Different from the first graph displayed, the points on this graph are not all lined up in a straight line. Here, the shape occurs to be a parabola. This graph however only includes the nonnegative numbers, meaning it is only half of a parabola Functions: y = ax2 + bx + c 1.
 Word count: 1642

Population trends. The aim of this investigation is to find out more about different functions that best model the population of China from 1950 to 1995.
Doing so has made the health care system be more effective which leads to more prevention of diseases and death. The investigation consists in finding a model that fits as close as possible the data given. This can be useful because the model can determinate how, if the parameters stay the same, the population numbers will be like in the future. However as parameters do change the models that are found will not be very precise to what the future could be, the past that is given can be modeled but that doesn't mean it will be similar to it.
 Word count: 4397

Math 20 Portfolio: Matrix
The variables are as follows: tn = triangular number (numbers of dots) n = the stage In order to modify the expression for the previous series of 100 consecutive positive numbers into the general formula for the triangular pattern, we can simply replace the number 100 within the expressing (50)(100+1) by n. The reason for which is because within the context of the series of 100 consecutive numbers, there is 100 terms and therefore the 100th stage. Also, as you can see the 50 within the expression is derived by dividing 100 by 2.
 Word count: 3011

The speed of Ada and Fay
Furthermore, after finding the time of Ada running 100 meters, I need to define the speed Ada runs in the experiment. To find the velocity, I will apply an equation that I use both in Physic and in Mathematic, which is . Therefore, by plugging the "s" and "t" values, then the result of the calculation will be the velocity. The calculation will be as follow, After calculation, I found that the velocity of Ada is 5.988 m/s. Due to further easier calculation; I decide to round up Ada's speed to one significant figure.
 Word count: 3314

Stellar Numbers
can see that in each term, as in the formation and the number of dots, the sum of the dots remain in a triangular pattern or shape. Furthermore, I can see that in each new term, one new row is added and the particular row has one more dot in the total number of dots than the previous dots. For example, in the first term, we see that there is one row only and in total, one dot in that row.
 Word count: 1877

Stellar numbers
Triangular Numbers (Total number of dots in a triangle) The first shapes that shall be considered are the triangular figures: When the values of the triangles (the number of dots) are input into a table: Stage of triangle Image of Triangular shape Counted number of dots 1 1 1 2 1 2 3 3 3 1 3 6 6 4 1 4 10 10 5 5 15 15 The variables will be defined the same for tables to do with triangluar numbers: n will be defined as the stage number of the triangle  as the nth stage of the triangle The variables will be defined the same for
 Word count: 2259

This paper attempts to analyze the formula modeling the motion of a freight elevator:
From t = 4 to t = 6, the velocity increases, which is parallel to the displacement graph where the elevator moves towards the ground from t = 4 and t = 6. From the graph of acceleration versus time, at first, the elevator speeds up at first from t = 0 and t =2. Comparing to the velocity graph, it can be seen that both acceleration and velocity are negative from t = 0 to t = 2. From t = 2, a(2)
 Word count: 1669

Math Portfolio Type 2
Perception is an interesting way of knowing, as sometimes it can become subjective without being noticed. "We are constantly deciding what to focus on, what to ignore, what is information and what is background noise." Once perception is subjective in its own nature, it cannot be the only tool to be used when it comes to judge a problem. In the old days, whenever people looked at the moon, all they saw was a spherical, white, and smooth planet. There were myths, fairy tales as well as superstitions involving the moon. For example, the Chinese and other Asian countries believed that there was a fairy living on the Moon, she will visit the Earth in the middle of August every year.
 Word count: 1556

Math Portfolio
The Volume of the water can be measured in Cubic Feet, and the Time in Hours. This gives us the Flow Rate in Cubic Feet Per Hour, however, a more standard unit has been derived which calculates the time in seconds, giving us the Flow Rate in Cubic Feet Per Second (cfs). Flow Rate of Nolichucky River between 27th October 2002 and 2nd November 2002. Time 0 6 12 18 24 30 36 42 48 54 60 66 72 Flow Rate (cfs) 440 450 480 570 680 800 980 1090 1520 1920 1670 1440 1380 Time 78 84 90 96 102 108 114 120 126 132 138 144 Flow Rate (cfs)
 Word count: 1839

This essay will examine theoretical and experimental probability in relation to the Korean card game called SutDa. First, a definition of probability and how it is used in general life will be examined. Each hand of SutDa provides the theore
Hence came to decision to make this as my extended essay topic. I will first introduce what is probability and the game called "SutDa". Everyone has different definition for probability and it depends on which perspective the people looks from, but generally, probability is the measure of how likely for an event to occur. Throughout this essay, these two games will be set under few conditions. These conditions are; 1. Number of players are 2 2. The deck is shuffled 10 times after the game in order to create a fair deck every round My research question becomes: What are the possibilities of winning in "SutDa" and how does the theoretical value compare with the experimental value?
 Word count: 5446

Statistics project. Comparing and analyzing the correlation of the number of novels read per week and the modal grade of T.I.S students
gender. Table of contents Page 1........... .......................................................Statement of task Page 2..................................................................Data collection procedure Page 36...............................................................Calculations (Statistics) Page 712..........................................................Calculations (Chisquare test) Page 1316....................................Calculations (Pearson's Correlation coefficient) Page 17..........................................Analysis and Conclusion Page 18............................................Reliability and Validity BOYS DATA Boys Number of books read mark 1 2 81100 2 1 4060 3 1 6180 4 2 6180 5 2 6180 6 5 4060 7 1 4060 8 2 4060 9 1 81100 10 6 6180 11 6 6180 12 6 6180 13 4 6180 141 2 6180 15 1 4060 16 2 4060 17 4 6180 18 1 4060 19 1 6180 20 1 below40 21 1 6180 23 1 6180 24 2 81100 25 1 4060 26 6 6180 27 1 6180 28 6 below 40 29 1 4060 30 6 6180 MARK AVERAGES (BOYS)
 Word count: 2736

Investigating Divisibility
= n (n + 1) (n  1) divisible by 3? When n = 1 n (n + 1) (n  1) = 1 (2) (0) = 0 As 0�3 = 0, P(n) is divisible by 3 when x = 3 and n = 1 Assume n = k is correct k (k + 1) (k  1) = k (k2  1) = k3  k = 3M (where M s any natural number) Then considering n = k + 1 (k + 1) {(k + 1) + 1} {(k + 1)  1} = (k + 1)
 Word count: 3562

Investigating graph of trigonometric function
If however we reduce the value, then the curve will vertically contract with an amplitude of the same value. Finally, if we inverse the sign of the amplitude for example we change 'a' into 'a', then the curve will reflect through the centreline. If we consider an infinitely extended graph, then we could say that the values of 'a' could be infinite apart from zero as the curve would vertically stretch or contract at any number apart from zero.
 Word count: 478

Parabola investigation. In this task, we will investigate the patterns in the intersections of parabola and the lines y =x and y=2x.
At first, we consider the parabola, the lines and. Again, we use the Graphmatica software to obtain four intersection points, repeat from step a to step c. Then label the intersections on the graph shown below. The xvalues of these intersections from the left to the right on the xaxis: > x1� 1.807 > x2� 2.268 > x3� 5.732 > x4� 7.193 Find the values of and and name them respectively SL and SR. SL = x2x1� 2.2681.807� 0.461 SR = x4x3� 7.1935.732� 1.461 Finally, calculate. Secondly, we consider the parabola, the lines and.
 Word count: 1415

Investigating Quadratic functions
A certain transformation is used for this graph from y=x� to become y=x�+k. It is a vertical translation since all three functions are the same except for their positions on the graph which is spread out vertically. Now, I will graph another set of functions of the family y=(xh) � where h is a constant term. From plotting this, I will obtain facts on how the value of h will have an effect on the position and shape of the graph.
 Word count: 1783