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Math IA - Logan's Logo

Extracts from this document...

Introduction

IBO INTERNAL ASSESMENT

Logan’s Logo

Mathematics SL Type II


Introduction

Logan has designed the logo at the right. The diagram shows a square which is divided into three regions by two curves. The logo is the shaded region between the two curves.image04.png

He wishes to find mathematical functions that model these curves.

In order to find these functions, we will need to overlay the logo on graph paper, so we can interpret data points to be able to plot them.

Take note that in the “modeling the data” in the next section, the logo was not resized, but set as transparent so that data points could be read.

Also, take into consideration the uncertainty of the measurements (± 0.25 units). For modeling purposes, the uncertainties are not included in the data calculations; however this should not be overlooked.

Modeling the Data

image05.pngimage14.pngimage00.png

image82.pngimage91.pngimage01.png

Note: Data tables and their graphs are included on the next page.


Top Curve image37.png:image05.pngimage14.png

In order to find a function to model the top curveimage06.png, there are various methods that we can use. One is to overlay the logo onto a set of axes and estimate points for the function. Once we obtain these points, we can then plot them onto a new set of axes. Judging from the logo itself, at first glance it appears that a sine function would fit the data. The sine function would have to undergo a series of transformations to eventually fit the curve. image91.png

Using the axes and logo depicted above, I estimated 13 points for image06.png and recorded them in the data table below:

NOTE: Due to the limited precision of the graph, I was only able to estimate to the nearest tenth. Maximum and minimum points have been shaded.

X

Y

-2.5

-1.0

-2.0

-2.5

-1.6

-2.8

-1.0

-2.4

-0.5

-1.3

0.0

0.0

0.5

1.5

1.0

2.6

1.5

3.4

1.8

3.5

2.0

3.4

2.5

2.5

2.6

1.9


...read more.

Middle

Now we just need to restrict the domain and range to the parameters we set at the beginning of the investigation. image102.pngThe final graph is shown at the left.

The small adjustments we made at the end exhibited the error in our calculations to find the curve of best fit. This can, in turn, be attributed to the large uncertainty in obtaining the actual data. Because I was only able to read points off of the graph, smallest of 1 unit, my precision was limited to one decimal place, and the uncertainty was quite large (±0.5 units). Another limitation was the thickness of the line in the original logo. Because it was so thick, it was also hard to determine where exactly the points lied. This increased the uncertainty of our measurement. Therefore, small adjustments to the curve were needed at the end to ensure the function best matched the data.


Bottom Curve image52.png:image05.png

In order to find a function to model the bottom curveimage104.png, we can also take the sinusoidal function approach. image91.pngimage103.png

Again, using the axes and logo depicted above, I estimated 13 points for image104.png and recorded them in the data table below:

NOTE: Due to the limited precision of the graph, I was only able to estimate to the nearest tenth. Maximum and minimum points have been shaded.

X

Y

-2.5

-3.1

-2.0

-3.5

-1.5

-3.4

-1.0

-2.6

-0.5

-1.8

0.0

-0.8

0.5

0.1

1.0

0.7

1.4

0.9

1.5

0.8

2.0

0.3

2.5

-1.0

2.6

-1.6


After determining these data points, I then plotted them onto a separate set of axes:

image105.pngimage103.png

From here, it is obvious that a sine function would fit the data. A sine function can be defined as:

image32.png,

Where a represents the amplitude of the sine curve (or vertical dilation); b is the horizontal dilation; c is the horizontal shift; d is the vertical shift; and x and y

...read more.

Conclusion

bHorizontal shift– cVertical shift– d

Numbers 1 and 4 both stretch or shrink the curve parallel to the x-axis, which means that the amplitude a and vertical shift d will only be affected by the manipulated range (factor ofimage31.png orimage33.png). On the other hand, numbers 2 and 3, the horizontal dilation and shift, will stretch or shrink the curve parallel to the y-axis. Again, this means that they will only be affected by the manipulated domain (factor ofimage27.pngorimage29.png).

Now we are able to apply the corresponding shrink factors to their appropriate variables in order to obtain a new curve for Logan’s business cards.

Top Curveimage37.png

The final equation for the top curve is given by the following equation:

image38.png with a period of 6.3.

Thus we must make the following changes to each of the variables. For a, the amplitude or vertical dilation factor:

image39.png

image40.png

For b, the horizontal dilation factor:

image41.png

image42.png

image43.png

image44.png

For c, the horizontal shift factor:

image45.png

image46.png

For d, the vertical shift factor:

image48.png

image49.png

Thus the final equation for the logo with dimensions compatible with that of a standard business card is: image50.png with parameters of: image30.pngandimage35.png.

The graph of the manipulated curve is shown below, on a 9cm × 5cm business card.

image51.png


Bottom Curveimage52.png

The final equation for the top curve is given by the following equation:

image53.png with a period of 6.0.

Thus we must make the following changes to each of the variables. For a, the amplitude or vertical dilation factor:

image54.png

image55.png

For b, the horizontal dilation factor:

image57.png

image58.png

image59.png

image60.png

For c, the horizontal shift factor:

image61.png

image62.png

For d, the vertical shift factor:

image63.png

image64.png

Thus the final equation for the logo with dimensions compatible with that of a standard business card is: image65.png with parameters of: image30.pngandimage35.png.

The graph of the manipulated curve is shown on the next page, on a 9cm × 5cm business card.

image67.png


The final logo on the business card is shown below. Note that the dimensions are in units which, shown through the calculations above, have been converted to ensure they match the ratio and proportion of a standard business card.

image70.pngimage68.pngimage69.png

 | Page

...read more.

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