# Logan's Logo

### Aims of this investigation:

* To find mathematical functions which model the upper and lower curves forming the logo

* To find the modified functions when printing the logo on T-shirts

* To find the modified functions when printing the logo on business cards

* To find the fraction of the area of the business cards occupied by the logo

### The following variables will be used in this investigation:

* f(x) à function f of x

* g(x) à function g of x

* x|-1, 1|à x is element of modulus between -1 and 1 (for instance)

* y|-1, 1|à y is element of modulus between -1, 1 (for instance)

* aà the amplitude

* bà the period, t

* cà the horizontal translation

* dà the vertical translation

I was given the following information:

Logan has designed the logo shown below. The diagram shows a square which is divided into three regions by the two curves. The logo is the shaded region between the two curves.

Logan wishes to develop mathematical functions which model these curves.

In order to find these functions I will have to find data points of the curve. I will do this by overlaying the given curves on graph paper. This will allow me to interpret the data points and hence plot the graph. The curve will be transparent so the data points can determined. It has to be noted, though, that since the measurements of diagram were done manually, there might be some wrong measurements (uncertainties). These lie between +0.25 and -0.25 cm. These uncertainties are not taken into consideration initially for modelling the functions but will be considered later on. The graph is shown on the next page.

Graph showing the two curves which enclose the logo (transparent) overlying graph paper

As you can see on the graph, I have labelled the curves f(x) and g(x). I will find the function of the top curve, f(x), first. I will do this by overlaying f(x) on graph paper and then read off twelve data points which will then be plotted on a different set of axes.

### 12 points of f(x) have been obtained (shown in the table):

x value

y value

-2.5

-1.0

-2.0

-2.5

-1.6

-2.8

-1.0

-2.4

-0.5

-1.3

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

In red are the maximum and minimum points.

It has to be noted that these data points are read off to the nearest tenth as the graph lacks accuracy.

These points have been plotted on a different graph (shown on the next page).

Graph showing the data points of f(x) (of the table on the previous page)

When looking at the graph, it becomes clear that the curve describes a sine function. This function is described by:

f(x) = asin |bx+c| +d

à a represents the amplitude

à b represents the period

à c represents the horizontal translation

à d represents the vertical translation

The curve has this function because the original sine curve has undergone a series of transformations resulting in its current shape. I will now find the values of a, b, c and d.

To find a (the amplitude), the maximum and minimum points of the curve have to found, the difference between them determined (to find the height of the curve), and then divided by two. According to my table, the maximum point is at (1.8, 3.5) and the minimum point is at (-1.6, -2.8). Since we are calculating the height (the y value), the lowest y value is subtracted of the highest y value. This means that the height of the curve is: 3.5- (-2.8) = 6.3. To find the amplitude, the height has to be divided by two: = 3.15.

However, as we can see on the graph, the curve initially has a negative slope (until (-1.6, -2.8)) which means that a has to be negative. Therefore, a = -3.15.

To find b, one has to determine the new period of the function. The period of a usual sine curve is 360° (or 2 radians). The equation for finding the period, t, is: t =.

This means that f(x) will have b cycles in 2. This can be found by finding the difference between the x-values of the maximum and minimum of the curve. I know that the maximum value is (1.8, 3.5) and that the minimum value is (-1.6, -2.8).

Rearranging the equation to find b gives me:

b =. Substituting t with 1.8- (-2.8) gives me:

b =. However, since this only gives me half the period, t has to be doubled to give me:

b =. Therefore, b =.

To find c, one has to note that f(x) = sin(x+ c) means that the curve is horizontally translated (the x value is subtracted by the value of c), while f(x) = sin(x- c) means that the values of x and c are added. Therefore, f(x) = sin(x+ c) corresponds to a translation to the left and f(x) = sin(x- c) to a translation to the right.

Using the origin of the curve (0, 0) I can deduce the horizontal translation of the original sine curve. When comparing the original sine curve to the translated curve, one can see that it has been translated to the left. Thus, the middle line of the height of the curve (which I have already found out as it is the amplitude, a). I can therefore graph y = 3.15 (the value of a) to find the middle line.

Graph showing the data points of f(x) and the line y = 3.15

However, as you can see on the graph, this is clearly not the middle line. If -2.8 (the minimum y-value) is added to 3.15, this gives me the new middle line (y = 0.35). The subsequent graph is shown below.

Graph showing the data points for f(x) and the line y = 0.35

y = 0.35

In general, a sine curve has the x-axis as the middle line and goes through the origin at (0, 0). When comparing this to the curve I have obtained I can see that this curve starts at (-3.0, 0.35). Therefore, c = 3.0.

To find d, one has to note that f(x) = sin(x) + d means that the curve is vertically translated (the value of d is added to the y value), while f(x) = sin(x) - d means that the values of y and d are added. Therefore, f(x) = sin(x) + d corresponds to an upwards translation, and f(x) = sin(x) - d to a downwards translation. To calculate d, the maximum value of y is added to the minimum value of y: 3.5+ (-2.8), and the result is then divided by two, in order to determine the new position of the x-axis. Therefore,

= 0.35.

d = 0.35.

Hence, I can conclude that

f(x) = -3.15 sin

I will now graph this curve on the same axes as my data points to see whether they are correct.

Graph showing f(x) and data points obtained previously

As you can see on the graph, the curve matches the data points closely, except in the area circled red. If I changed the period, t, (the variable b) this might change. If I change b from 0.92 to 1.01 ( rounded to the nearest hundredth). This gives me the following curve (on the next page).

Graph showing f(x) and altered f(x) (after increasing b)

As you can see on the graph, the curve f(x) = -3.15 sin(1.01x + 3.0) + 0.35 fits the data points better. I will hence use this as my final equation for f(x):

f(x) = -3.15 sin(1.01x + 3.0) + 0.35

To produce the final graph, I have to cross-reference with the parameters I determined in the beginning:

x|-2.5, 2.7|

y|-3.0, 3.5|

Therefore, my final graph is:

I will now find g(x). I will follow the same method as for f(x).

Again, I have produced a results table for the data points of g(x) based on the first graph in the beginning of this investigation.

x value

y value

-2.5

-3.1

-2.0

-3.5

-1.5

-3.4

-1.0

-2.6

-0.5

-1.8

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

In red are the maximum and minimum points.

It has to be noted that these data points are read off to the nearest tenth as the graph lacks accuracy.

These points have been plotted on a different graph (shown below).

Graph showing the data points of g(x) (of the table above)

Similar to the function of f(x), this curve seems to present a sine function:

f(x) = asin |bx+c| +d

à a represents the amplitude

à b represents the period

à c represents the horizontal translation

à d represents the vertical translation

The curve has this function because the original sine curve has undergone a series of transformations resulting in its current shape. I will now find the values of a, b, c and d using the same method I used for f(x).

To find a, the difference between the maximum and minimum y-points of the curve have to be found. These maximum and minimum points are (1.4, 0.9) and (-2.0, -3.5).

Therefore, 0.9-(-3.5) = 4.4. This gives me the height of the curve. To determine the amplitude, a, this has to be divided by two:

= 2.2. Similar to f(x), the curve g(x) starts with a negative slope which means that the amplitude, a, has to be negative. Therefore, a = -2.2.

To find the b (the period t), the equation to find t has to be considered: t =.

This means that g(x) will have b cycles in 2. This can be found by finding the difference between the x-values of the maximum and minimum of the curve. I know that the maximum value is (1.4, 0.9) and that the minimum value is (-2.0, -3.5).

Rearranging the equation to find b gives me:

b =. Substituting t with 1.4- (-2.0) gives me:

b =. However, since this only gives me half the period, t has to be doubled to give me:

b =. Therefore, b =.

To find c, one has to note that g(x) = sin(x+ c) means that the curve is horizontally translated (the x value is subtracted by the value of c), while g(x) = sin(x- c) means that the values of x and c are added. Therefore, g(x) = sin(x+ c) corresponds to a translation to the left and g(x) = sin(x- c) to a translation to the right.

Using the origin of the curve (0, 0) I can deduce the horizontal translation of the original sine curve. When comparing the original sine curve to the translated curve, one can see that it has been translated to the left. Thus, the middle line of the height of the curve (which I have already found out as it is the amplitude, a). I can therefore graph y = 2.2 (the value of the height divided by two, which is a) to find the middle line. The graph is shown on the following page.

Graph showing the data points of g(x) and the line y = 2.2

However, as you can see on the graph, this is clearly not the middle line. If -3.5 (the minimum y-value) is added to 2.2, this gives me the new middle line (y = -1.3). The subsequent graph is shown below.

Graph showing the data points for f(x) and the line y = -1.3

In general, a sine curve has the x-axis as the middle line and goes through the origin at (0, 0). When comparing this to the curve I have obtained I can see that this curve starts at (-3.4, -1.3). Therefore, c = 3.4.

To find d, one has to note that g(x) = sin(x) + d means that the curve is vertically translated (the value of d is added to the y value), while g(x) = sin(x) - d means that the values of y and d are added. Therefore, g(x) = sin(x) + d corresponds to an upwards translation, and g(x) = sin(x) - d to a downwards translation. To calculate d, the maximum value of y is added to the minimum value of y: 0.9+ (-3.5), and the result is then divided by two, in order to determine the new position of the x-axis. Therefore,

= -1.3.

d = -1.3.

Hence, I can conclude that

g(x) = -2.2 sin

I will now graph this curve on the same axes as my data points to see whether they are correct.

Graph showing g(x) and data points obtained previously

As you can see on the graph, the curve matches the data points closely, except in the area circled blue. If I changed the period, t, (the variable b) this might change. If I change b from 0.92 to 1.01 ( rounded to the nearest hundredth). This gives me the following curve (shown below).

Graph showing g(x) and altered g(x) (after increasing b)

As you can see on the graph, the curve g(x) = -2.2 sin(1.01x + 3.4) – 1.3 fits the data points better. I will hence use this as my final equation for f(x):

g(x) = -2.2 sin(1.01x + 3.4) – 1.3

To produce the final graph, I have to cross-reference with the parameters I determined in the beginning:

x|-2.5, 2.7|

y|-3.6, 0.75|

Therefore, my final graph is:

In general, it has to be noted that there is a large uncertainty (0.5) in getting the data which probably caused the errors at the end of the calculations for both f(x) and g(x). This was because I had to determine the points visually. Also, the line of the curve is very thick which means that the points is deduced could lack accuracy. However, through changing b in the end of the calculations for both curves, this could be reduced.

Now, Logan wishes to print T-shirts with the logo on the back. For this purpose, she must double the dimensions of the logo. This means that I have to modify the functions I have just found for the two curves.

Since the dimensions of the logo are doubled, the width and the length are multiplied by two. Therefore, a, c, d must be multiplied by two and b has to be divided by two (since it is the period and the number of cycles increases the bigger the area is).

a, the amplitude, has to be multiplied by two which means that all y-values double. This results in a vertical stretch of a factor of 2.

b, the period, has to be divided by two.

c, the horizontal translation, has to be multiplied by two since the dimensions of the logo are doubled so the horizontal translation has to be doubled as well.

d, the vertical translation has to be doubled for the same reason as c has to be doubled.

The parameters also have to be doubled which gives me:

### Original: New:

x|-2.5, 2.7| x|-5, 5.4|

y|-4, 4| y|-8, 8|

This gives me the following graph (shown on the next page).

Graph showing f(x) and g(x) and the new functions for the dimensions of the T-shirt

f(x) and g(x) are the original curves of the logo before changing the dimensions for the T-shirts. f2(x) and g2(x) are the curves after the dimensions have been doubled.

Now, Logan wishes to print business cards. A standard business card is 9cm by 5cm. This means that I have to modify the functions I have deduced in order for it to fit on the business card exactly.

As a first step, the domain and range have to be modified. The dimensions right now are 10cm 9.4cm and I have the following parameters:

x|-2.5, 2.7| à domain

y|-4, 4| à range

As I can see from the domain, the length of the logo is 5.2 units at this moment. This has to be seen relative to 10cm of the business card. As I can see from the range, the width of the logo is 8 units at this moment. This has to be seen relative to the 9.4cm of the business card.

Therefore, the domain is and the range is.

The dimensions are 9cm by 5cm. Therefore, to find the new domain:

Hence x equals to 4.68. This means that the total length (of the domain) is 4.68. I have left the minimum x (-2.5) and added 4.68 to it which gives me the new maximum x (2.18). Therefore, the new domain is: x|-2.5, 2.18|.

To find the range:

=.

Hence y equals to 4.255. This means that the total length (of the range) is 4.255. I have left the minimum y (-4) and added 4.255 to it which gives me the new maximum y (0.255).

Therefore, the new range is: y|-4, 0.255|

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