• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Investigate the mathematical functions which model these curves and determine Logans logo

Extracts from this document...

Introduction

In this assignment, I will investigate the mathematical functions which model these curves and determine Logan’s logo.

For this, I will use an appropriate set of axis with a scale of 1 centimetre per unit and record a number of different data points on the curves which will then create model functions.

Logan’s logo represents two curves: the lower curve and the upper curve. The variables used are the horizontal-axis and the vertical-axis.

The measured numbers of data points for the lower and upper curve are represented in this table.

1. The various numbers of data points from the curve are entered in a table as shown below. L1 being the horizontal-axis, L2 the lower curve and L3 the upper curve.

image07.png

The window parameters for the graph are then programmed for both the upper and the lower curve to respect the scale on Logan’s logo:

image08.png

Then a scatterplot is obtained for both curves after putting the points of each curve into a table.

image17.png

Graph of the lower curve:

image22.png

...read more.

Middle

Using the sinusoidal regression formula the values of a, b, c and d for the upper sine curve are found:

image11.png

Following the sine formula, the letters are replaced with the values found with the sinusoidal regression formula in the function plot to then create a graph for each curve.

image12.png

The graph of the lower sine curve resembles this:

image13.png

The same method is replicated for the second upper sine curve:

image14.png

The graph of the upper sine graph is represented as the following:

image15.png

The amplitude of the upper sine curve is calculated with the following formula: image16.png

However, the sine curves of Logan’s logo carry some limitations.

First of all, it is not certain that this curve follows the sinusoidal behaviour. It is only a possible hypothesis. Furthermore, the values are not extremely precise therefore the results of the sine equations are not very accurate. Thirdly, as it was graphically impossible to obtain the correlation coefficient (r),

...read more.

Conclusion

image09.png5.70

The amplitude is measured with the following formula:

Amplitude=image16.png=

image23.png

ower limit=0

Upper limit=8

3. For the business card, we must multiply the amplitude by 5/8 and to find the area between two curves, we find the area under the curve which is above and subtract the area of the curve which is below.

The lower limit is 0 and the upper one is 8.  (screenshot 2nd calc , 7)

In order to find the area between the two curves, we have to substract the area of the lower curve to the area of the upper curve.

image18.png32.13-20.15image24.png

The business card’s area is 45cm2. The amplitude of the two sine functions must be multiplied by image25.png to reduce the height (y-axis) to 5 cm

Logan’s logo occupies image26.png of the business card.

Pour la 1ère function A has to be multiplied by 5/8 and d which affects vertical translation  ce qui donne 1.27 et d= 2.42*5/8=1.51

Pour la 2ème function a (amplitude) and the d by 5/8 2.93 * 5/8=1.89. et d=4.03*5/8=2.52

Screenshots of business-card format, sine functions :

Then calculate the integrals and area

Sine function 1: aimage21.png

image27.png for the lower curve

For the upper curve f(x)=2axsin(bx+c)d

image28.png cm 2 for the upper curve.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. LACSAP's Functions

    of the new triangle as it's the only diagonal row that appears consistent with Pascal's triangle. n (Row number)

  2. Math IA - Logan's Logo

    points on the curve, which are (1.8, 3.5) and (-1.6, -2.8) respectively. When we find the amplitude of a sine curve, we can disregard the x-values, as they do not tell us anything about the height of the curve. Now that we've found the highest and lowest points of the

  1. Derivative of Sine Functions

    - - - 0 -1.00 -0.707 0 0.707 1.00 0.707 0 -0.707 -1.00 X 2 -0.707 0 0.707 1.00 0.707 0 -0.707 -1.00 The scatter plots of values in the table is sketched below: (joined by a smooth curve) gradient of f(x)

  2. Finding Functions to Model Population trends in China

    happen because from the graph, we could see that the slope of the graph is starting to change in year 1995, which may indicate the decrease of the population.

  1. Mathematics Higher Level Internal Assessment Investigating the Sin Curve

    From all the graphs, discussions and examples it is clearly seen that when the sine curve is written in the form we can get a lot about the graph. The represents the amplitude of the curve, the period of the graph is written by whereas the values of and translate the graph horizontally and vertically respectively.

  2. An investigation of different functions that best model the population of China.

    These are as follows: K= 3230.36868 L=4.77937 M=0.02431256 Therefore: Having calculated the population using the researcher's model, we can see that the table looks like this. Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 Population in Millions (Original)

  1. Population trends. The aim of this investigation is to find out more about different ...

    that there is no way that this can be the model, however the graph can still be manipulated to find a similar model. Again the in the equation is relevant because it was the population at the year where the data given starts.

  2. A logistic model

    The graph is asymptotic approaching 6.0x104 fish despite the years 19 and 20 showing data rounded off to 6.0x104. 4. The biologist speculates that the initial growth rate may vary considerably. Following the process above, find new logistic function models for un+1 using the initial growth rates r=2, 2.3 and 2.5.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work