Firas A. IB1-A

Internal AssessmentTrigonometric functions

Type I - mathematical investigation

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Firas A. IB1-A

1.

a) y = sin xb) y = sin x + 2c) y = sin x – 3

Figure 1.1 displays the graphs of the functions y = sin x, y = sin x+2 and y = sin x-3 in one Cartesian coordinate system. Looking at the different graphs depicted in the below fashion, one firstly recognizes that the three graphs follow exactly the same pattern for . Furthermore, they appear to be parallel to each other. Upon closer examination of the different functions’ numerical components, one realizes that the only difference between the underlying functions of the graphs is the factor following the “sin x” part of the function. (+2 in the case of y = sin x+2, 0 in case of y = sin x and -3 in case of y = sin x-3). As can be further inferred from the below observation, the sign of the factor seems to indicate the direction of the graphs upward or downward movement.

Based on these observations the following generalization can be made in this case:

If y = sin x is replaced by y = sin x + c the graph is moved upwards by c units if c > 0 (downwards if c is negative).

The validity of this general rule can be easily shown by providing two more examples (see figure 1.2). Here again one should realize that the graph of the function y = sin x+1.5 is obtained by moving the graph of y = sin x upwards by 1.5 units. Similarly the graph of y = sin x-1.5 is obtained by moving the graph of y = sin x downwards by 1.5 units.

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Firas A. IB1-A

Figure 1.1: Upward and downward movements of y = sin x

Figure 1.2: examples to support first generalization

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Firas A. IB1-A

2.

a) y = cos xb) y = 3 cos xc) y = ½ cos xd) y = -2 cos x

For the sake of clarity in this task each of the above graphs is compared separately against the graph of the function y = cos x. Looking at figure 2.1 it can be seen that the factor 3 in front of the “cos x” component has caused the graph to appear steeper. Furthermore, it seems as if the factor 3 stretches the graph of y = cos x in a vertical direction.

Remarkably the graph of y = 0.5 cos x behaves exactly the opposite way when compared to the previous example of y = 3 cos x. In fact, the factor 0.5 seems to have caused a horizontal stretch to the graph of y = cos x, as opposed to the factor 3 causing a vertical stretch. The graph of the function y = -2 cos x clearly seems to have been obtained by reflecting y = cos x about the x axis. Furthermore the graph of y = -2 cos x appears to be vertically stretched.

Figure 2.1: comparison between the graphs

Based on the observations made in task two one can formulate the following generalization: if y = cos x is replaced y = c cos x, the graph is stretched vertically if c > 1 and vertically compressed in y-direction if the absolute value of c lies between zero and one. Additionally, if the factor c has a negative sign the graph will be reflected about the x axis.

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Firas A. IB1-A

To further substantiate the generalizations made, the following figure provides examples of graph types similar to the ones in task two.

Figure 2.4: Additional examples

3.

a) The graph of y = 3 sin x – 2 can be obtained from y = sin x by shifting the graph of y = sin x according to the rules that have been established so far. The first step in this process is to stretch the graph vertically by the factor 3. Next, the obtained graph y = 3 sin x is moved downwards by two units, arriving at the function y = 3 sin x -2. Figure 3.1 further illustrates this approach, starting from y = sin x, moving to y = sin x -2 (dotted graph) and finally arriving at y = 3 sin x – 2.

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Firas A. IB1-A

Figure 3.1: two step process to arrive at y = 3 sin x -2

b)

In this case the similar approach of the previous task can be applied to obtain the graph of y = 2 - cos x from the graph of y = cos x. To avoid confusion and illustrate the similarity between the previous case and this one, y = 2 – cos x is rearranged to y = - cos x + 2. The first step towards obtaining y = - cos x + 2 from y = cos x is reflecting the latter graph about the x axis to obtain y = - cos x (see dotted graph in figure 3.2). This new graph is raised by two units, obtaining the graph y = 2 – cos x.

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Firas A. IB1-A

Figure 3.2: obtaining y = 2 – cos x from y = cos x

4.

Although the figure below might initially confuse the reader, it actually illustrates an interesting phenomenon associated with positioning the factor c in between the “cos” and x part of the function (e.g. the factor 2 in the function y = sin 2x. As the figures shows, the higher the factor c the stronger becomes the impression that the graph is being compressed. If the graph of y = sin x is compared to a sheet of paper, then increasing the factor c in this case would mean increasing pressure on both ends of the paper, causing it to look more compressed. This development can be clearly seen when comparing the graphs of y = sin 2x, y = sin 3x and y = sin 4x to each other. The latter seems to be the most horizontally compressed one. Besides the graphs with the higher factor show considerably more wave-like fluctuations. Considering the remaining graph y = sin 0.5x, its behaviour seems to be at odds with the other described graphs. In fact when comparing it to y = sin x, the graph of y = sin 0.5x appears to be rather stretched out.

Based on the observations above one might carefully formulate the following generalization: if y = sin x is replaced by y = sin c x, the graph is stretched horizontally in x direction if the absolute value of c lies between 0 and 1 (as in the case of y = sin 0.5x). Conversely, the graph is compressed horizontally if c > 0.

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Firas A. IB1-A

Figure 4.1: compressing y = sin x with y = sin c

Please have a look at the separate paper with all of the four graphs.

5.

Comparing the graph of the function y = cos x with y = cos (x-30°) one can notice that y = cos (x – 30º) seems to have been obtained from y = cos x by simply moving the graph rightwards by 30°. Similarly one can explain the way to arrive at y = cos (x – 60º) and y = cos (x – 90º) respectively based on the same pattern. Hence follows the generalization:

If y = cos x is replaced by y = cos (x + c) the graph is moved c units to the right if c < 0. It is also save to assume that for c > 0 the graph will move to the left.

Figure 5.1: moving y = cos x rightwards

To corroborate the above statement the following figure provides further examples of moving the graph leftwards and rightwards.

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Firas A. IB1-A

Figure 5.2: further examples of rightward and leftward movement

6.

Expanding the above developed concepts and ideas to the graph y = sin (2x – 60°) is again nothing but a combined application of the different rules of shifting graphs states above. Specifically in this case y = sin x is first compressed in x direction (refer to the paper sheet example) by the factor 2. Subsequently this graph is then moved rightwards by 60°, obtaining the graph y = sin (2x – 60°).

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Firas A. IB1-A

Figure 6.1: the whole shifting process is being illustrated below in this figure

7.

It is generally possible to obtain graphs like y = 2 cos (2x + 30°) – 2 from y = cos x following the process of systematic shifting the graph of y = cos x. Firstly y = cos x is replaced by y = 2 cos x causing the graph to be stretched in a vertical dimension. Secondly this graph is then moved horizontally compressed in x direction by the factor 2. After this the graph is moved 30° to the left. Finally this new graph is moved downwards by two units, arriving at the desired graph of the function y = 2 cos (2x + 30°) -2. Figure 7.1 illustrates the process just described. For clarity reasons the figure only shows the first and the final function, thereby omitting the intermediate steps. However this should not be of concern since the basic approach of shifting the graph is essentially similar to the previous illustrates. Therefore it is believed that no further examples are necessary at this point.

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Firas A. IB1-A

Figure 7.1: example integrating all previously developed ideas in one single graph!!

8.

So far, the discussion has been limited to trigonometric functions. However, it can be proven with great certainty that the set of rules concerning the shifting of graphs developed in this context can also be applied to other types of functions. Consider for example the quadratic function of the form y = - (x + 2) ^2 + 2. This seemingly complicated graph can be easily obtained from the graph of the parabola y = x^2, using the following steps. First y = x^2 is reflected about the x axis to obtain the graph of y = - x^2. This graph is then moved to units to the left, which results in the graph of y = - (x + 2) ^2. Finally, this new graph is raised by 2 units and the graph obtained is shown in figure 8.1.

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Firas A. IB1-A

Figure 8.1: applying the concepts of shifting to quadratic functions

General conclusion

This mathematical investigation has provided various interesting applications and generalizations regarding the properties of trigonometric functions. During the investigation the technique of shifting graphs of trigonometric functions has been examined from different perspectives, ultimately leading to a complete framework which can be generally applied to all various kinds of functions. The whole idea developed in this investigation can be summarized by the following mathematical expression: y = - a f(b x + c) + d, where the variables have the following effects:

Negative sign in front of “a” variable: reflects function about the x axis. “a”: determines the vertical stretch of the graph in y direction. “b”: determines the extend to which the graph is compressed in x direction. “c”: determines how many units the graph is moved to the left (when “c” is positive)

or right (when “c” is negative). “d”: determines how many units the graph is moved downwards or upwards. Positive

“d” implies upward movement while negative “d” implies downward movement.

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