sine waves transformations

1. 1
Math Project
What is the effect of changing the
transformations on a sine wave in the form of:
y=asin(bx+c)+d?
Candidate number:
Candidate name: Ahmad Hasan
Supervisor: Pro. Mohammed Saleem

2. 2
Contents
Introduction:
Statement of the task and the record of action 3
Measurements 4
Mathematical processes 5
Interpretation of results 13

3. 3
Introduction:
Statement of the task:
Sine and Cosine waves “commonly referred to as sinusoid “are mathematical functions
that occur in many fields of science such as mathematics, physics, and electrical
engineering. These waves are very unique because they retain their shape when added
to another sine wave which has the same frequency and arbitrary phase. This project
will study the effect of making change in the transformations of a certain wave. These
waves occur in different locations in the earth such as in the ocean, sound, and light
waves.
The transformations of a certain function are a, b, c, d written in the form:
y=asin(bx+c)+d, each letter refers to a certain transformation of the function. a which
represents the amplitude which is half the distance between the maximum and the
minimum values of the functions. The letter b the period of the function is an interval
that the function has a complete single cycle of the curve. The phase which is
represented by the letter c is the horizontal shift of the function; the vertical shift is
represented by the letter d.
Record of action:
 Obtain a sine equation in the form of y=asin(bx+c)+d , and graph it showing all the
translations on the diagram.
 Draw tables by which each of the transformations is referred to as an integer (x).
 Calculate the value of x after increasing and decreasing the value of x in each
transformation’s table.
 Draw a linear graph for each table, and write an explanation of what is the change
that occurred on the transformation after the increase or decrease where you
compare the increase with the decrease, for example what was the effect of
increasing the value of the period.
 Give comments on the results obtained from the project.
 Since sine waves occur in light, sounds, and ocean waves, find a certain conclusion
about what could be the effect of changing the transformations on one of these
three waves.

4. 4
Measurements:
Equation:
Y=sin(x)
Table 1:
x 0 30 45 60 90 120 135 150 180
Sin(x) 0.0 0.5 0.7071 8660 1.0 0.8660 0.7071 0.5 0.0
Graph 1:
This is a graph of the function y=sin(x) where x is from -270 until 810, and it shows on
the y axis the values of sin(x):
From this graph may conclusions are acquired including:
 The maximum value of Y is 1, and the minimum value is -1.
 The period= 360o
 The function is periodic which means that it repeats itself.
From this function the translations which were mentioned in the statement of the task
can be easily seen in the following sketch the translations are shown:
Sketch 1:
This sketch shows the translations on the sine wave graph:

5. 5
Mathematical processes:
Amplitude (a):
In the following tables the amplitude is going to be doubled by multiplying it with 2, and
halved by multiplying it with half, then the value of the amplitude is going to be calculated
at different values of x:
x 0 30 45 60 90 120 135 150 180
2sin(x) 0 1 1.4142 1.7321 2 1.7321 1.5321 1 0
x 0 30 45 60 90 120 135 150 180
1/2sin(x) 0 0.25 0.3536 0.4330 0.5 0.4330 0.3536 0.25 0
Now I am going to display the table of sin(x) so we can have an idea about what happened
to the values of sin(x) when it was multiplied by 2 and ½:
x 0 30 45 60 90 120 135 150 180
Sin(x) 0.0 0.5 0.7071 8660 1.0 0.8660 0.7071 0.5 0.0
Graph 2:
The following graph illustrates the three tables of sin(x), 2sin(x), and 1/2sin(x) where sin(x) is
represented in black, 2sin(x) is represented in green, and 1/2sin(x) is plotted in blue:
2sin(x)
Amplitude= |max-min value|/2
= |2--2|/2
=2
Period=360
Phase shift=0
vertical shift=0

6. 6
f(x)=1/2sin(x):
Amplitude=|1/2--1/2|/2 = ½
Period=360
Phase shift=0
Vertical shift=0
The Period (2∏/|b|):
In the following tables the period 2∏/|b| in the equation y=asin (bx+c)+d is going to be
increase (multiplied by 2) and decreases (multiplied by half):
x 0 30 45 60 90 120 135 150 180
Sin(2x) 0 0.8660 1 0.8660 0 8660 -1 -0.8660 0
x 0 30 45 60 90 120 135 150 180
Sin(1/2x) 0 0.2588 0.3827 0.5 0.7071 0.8660 0.9239 0.9659 1
Let’s remember the table when of sin(x) when it wasn’t multiplied by any number:
x 0 30 45 60 90 120 135 150 180
Sin(x) 0.0 0.5 0.7071 8660 1.0 0.8660 0.7071 0.5 0.0
Graph 3:
This diagram is a graph of the 3 tables of sin(x), sin(2x), and sin(1/2x) where sin(x) is plotted
in the black color, sin(2x) in green, and sin(1/2x) is represented in blue:
F(x)=Sin(2x):
Amplitude=1
Period= 360/2= 180
Phase shift=0
Horizontal shift=0

7. 7
F(x)=Sin(1/2x):
Amplitude=1
Period=360/0.5= 720
Phase shift=0
Horizontal shift=0
The phase shift (-c/b):
The following tables are going to investigate the effect of changing the phase shift which is
equal to –c/b in the equation y=asin(bx+c)+d assuming that b>0. One positive number, and
one negative number is going to be added to the equation, and the value of sin(x) is going to
be obtained:
x 0 30 45 60 90 120 135 150 180
Sin(x+45) 0.7071 0.9659 1 0.9659 0.7071 0.2588 0 -0.2588 -0.7071
X 0 30 45 60 90 120 135 150 180
Sin(x-60) -0.8860 -0.5 -0.2588 0 0.5 0.8660 0.9659 1 0.8660
The table for the values of sin(x) is:
x 0 30 45 60 90 120 135 150 180
Sin(x) 0.0 0.5 0.7071 0.8660 1.0 0.8660 0.7071 0.5 0.0
Graph 4(a):
This graph shows a plotting of sin(x) and sin (x+45) on the same diagram where sin(x) is
represented in black, and sin(x+45) is plotted in green:

8. 8
Graph 4(b):
This graph shows a plotting of sin(x) and sin(x-60) where sin(x) is plotted in black and
sin(x-60) is represented in blue:
Graph 4(c):
This graph shows a plotting of all three functions (f(x)=sin(x), y=sin(x+45), and f(x)=sin(x-60)
Where sin(x) is represented in black, sin(x+45) is plotted in green and sin(x-60) in blue:
F(x)=Sin(x+45):
Amplitude=1
Period=360
Phase shift= -45/2(1)
= 45 unit to the negative side (left)
Vertical shift=0

9. 9
F(x)=Sin(x-60):
Amplitude=1
Period=360
Phase shift=-(-60/1)
= 60 unit to the positive side (right)
The vertical shift (d):
In the following tables the vertical shift d in the equation y=asin(bx+c)+d is going to be
changed by adding 3 and subtracting 4 from the equation of sin(x):
x 0 30 45 60 90 120 135 150 180
Sin(x)+3 3 3.5 3.7071 3.8660 4 3.8660 3.7071 3.5 3
x 0 30 45 60 90 120 135 150 180
Sin(x)-4 -4 -3.5 -3. 2929 -3.1340 -3 -3.1340 -3.2929 -3.5 -4
Graph 5:
This graph shows the effect of adding 3 to sin(x) (in green) and subtracting 4 to sin(x) (blue)
and the original sin(x) graph in the black colour:
F(x)=Sin(x)+3:
Amplitude=1
Period=360
Phase shift=0
Vertical shift= 3 units upward
F(x)=Sin(x)-4:
Amplitude=1
Period=360
Phase shift=0
Vertical shift= 4 units downward

10. 10
Now we are going to see the transformation on the function f(x)= sin (x)+cos(x):
X 0 30 45 60 90 120 135 150 180
Sin(x)+cos(x) 1 1.3660 1.4142 1.3660 1 0.3660 0 0.3660 -1
Graph 6:
This graph illustrates the change occurred when cos(x) was added to the function of sin(x)
and it also shows the original function of sin(x):
F(x)=sin(x)+cos(x):
Period= 360
Amplitude= |1.4142-- 1.4142|/2
=1.4142= √2
Phase shift=45 (from the graph)
Therefore the graph is shifted 45 units towards the negative side (left)
Vertical shift=0
Therefore we can say the sin(x)+cos(x)= √2sin(x+45)
There is one more thing that we are going to apply to the function of sin(x) which is
multiplying it with a negative number which is -1:
X 0 30 45 60 90 120 135 150 180
-sin(x) 0.0 -0.5 -0.7071 -8660 -1.0 -0.8660 -0.7071 -0.5 0.0

11. 11
Graph 7:
This graph shows the effect of multiplying sin(x) with negative one by plotting the function
of sin(x) in black and -sin(x) in green:
Amplitude=1
Period=360
Phase shift=0
Horizontal shift=0
After seeing the effect of multiplying sin(x) with a negative number we are going to see the
how the function f(x-90)= sin(x-90)+cos(x-90) differs from f(x)= sin(x)+cos(x):
x 0 30 45 60 90 120 135 150 180
Sin(x-90+cos(x-90) -1 0.3660 0 0.3660 1 1.3660 1.4142 1.3660 1
Graph 8 this graph shows the functions f(x)=sin(x) (in black), f(x-90)= sin(x-90)+cos(x-90)
(plotted in blue) f(x)= sin(x)+cos(x) (plotted in green):

12. 12
F(x-90)=sin(x-90)+cos(x-90)
Amplitude= |1.4142--1.4142|/2
= √2
Period=360
Phase shift= -45 (45 units towards the positive side (right))(from the graph)
Vertical shift=0
Finally the function f(x-90)=sin(x-90)+cos(x-90) is going to be multiplied by 2 therefore
2(f(x-90) = 2(sin(x-90)+cos(x-90)), The following table is going to show the values of of
2(f(x-90) at different values of x:
X 0 30 45 60 90 120 135 150 180
2(sin(x-90)+cos(x-
90)
-2 -0.7321 0 0.7321 2 2.7321 2.8284 2.7321 2
Graph 9:
The following graph plots the functions of f(x-90)=sin(x-90)+cos(x-90) (in green) and
2(f(x-90) = 2(sin(x-90)+cos(x-90)) (in gold):
Period= 360 (from the graph)
Amplitude= |2.8284--2.8284|/2
=2.8284
Period=360
Phase shift= -45 (45 units towards the positive side (right))(from the graph)
Vertical shift=0 (from the graph)

13. 13
Interpretation of results:
Amplitude (a):
As what was written before the amplitude is equal to half the maximum distance between
the greatest and the least value in a certain function, however in a simple sine function the
amplitude can be acquired as it is the coefficient of sin(x), in this project the amplitude was
increase by being multiplied by 2, and decreased by multiplying it by half. When the
amplitude was multiplied by 2 the graph stretches the graph of 2sin(x) was vertically
dialated on the positive and negative axis. However when Sin(x) was multiplied by half the
shrinks its graph was compressed, however it should be noted that changing the amplitude
doesn’t change the value of the period, phase, or vertical shift. The graph of 2Sin(x) has a
significantly larger amplitude which is 2 than 1/2Sin(x) who has the amplitude of 1/2
therefore the graph of 2sinx is longer in a vertical form from ½ sin(x).
Period ((2∏/|b|)):
The period of a function is a complete single cycle of the sine function when was multiplied
by 1/2 the period increases to 720, which means that it took the function 720 units to
complete a single cycle, however when the period was multiplied by 2 the period decreases
to 180 which means that the full cycle of sin(2x) is more frequent than the period of
sin(1/2x), we can also say that one period of sin(1/2x) contains for complete periods of
sin(2x) and this can be seen easy from the graph. In other words If the b is bigger than 1, the
period will shrink or stretch horizontally. If the number B is smaller than 1, the period will stretch It
should be noted that changing the period of a certain function doesn’t affect its other
transitions including the amplitude, phase shift and the vertical shift.
The Phase shift(-c/b):
The phase shift is the horizontal shift of a function, however it should be noted that
increasing the value of c in the equation y=asinb(x-c)+d shifts the function towards the
negative side (left) and this was observed graph 4(a) where the function of sin(x+45) was
moved 45 units towards the left side however in the function of sin(x-60) the effect of
subtracting 60 from the function (decreasing the phase shift) was a move towards the
positive side (right) in 60 units, however both function have the same period, amplitude
vertical shift, and this can be observed from the graph.
The vertical shift (d):
The vertical shift is the move of the function either upward or downward. Increasing d in the
function y=asin(bx+c)+d will move the function d units upwards, however decreasing the
value of d will shift the function downward, all these things were observed in graph 5 where
adding 3 to sin(x) sin(x)+3 moved the graph of sin(x) upward by 3 units however when we
subtracted 4 from the function of sin(x)  sin(x)-4 the graph shifted 4 units downwards.
Therefore the graph of sin(x)+3 is above sin(x)-4 by 6 units. However both functions have
identical amplitudes, periods and phase shift.

14. 14
F(x)=Sin(x)+Cos(x):
When the function of sin(x) was added to cos(x) the resultant was a diffrent graph with
different features from the graph of sin(x), f(x)=sin(x)+cos(x) had a larger amplitude which
had the value of √2, and its phase changed to 45 which means that it was moved 45 units
towards the negative axis, the graph was vertically dilated. However the period and the
phase shift stayed the same where the period remained 360 and the graph wasn’t shifted
neither upward nor downwards.
F(x)= -sin(x):
When we multiplied sin(x) by a negative number the function was flipped or reflected, this
happens because the values of y will be reversed in sign from those that were calculated
from the sin(x) graph, however there is no effect of this transformation on any of the
translation, graph 6 shows that all the translations remained constant however the –sin(x)
function can be equal to the functions sin(x+180) sin(x-180) and many other forms, this is
because of the periodic nature of the functions there for we can also say that the phase shift
of this function is ±180.
F(x-90)= sin(x-90)+cos(x-90):
When f(x) was changed into f(x-90) the function was shifted 45 units towards the positive
axis, however the other translations remained the same as the translations in the function
f(x)=sin(x)+cos(x).
2f(x-90)= 2(sin(x)+cos(x)):
When f(x-90) was multiplied by 2 the most important change that was observed is that the
amplitude increased however it wasn’t exactly equal to 2 it was 2.8284 this is because of the
addition of cos(x) to the function, the graph was vertically dialated, however all the
remaining translation stayed the same as in the function f(x-90)= sin(x-90)+cos(x-90) and
this can be observed from graph 9.
Connection with sound waves:
As written in the introduction sine waves occur in sound waves in physics it is known that
the period is inversely proportional to the frequency (as the period increases the frequency
decreases), however increasing the frequency of a sine wave will lower the pitch of the
voice making it thicker and deeper. Therefore we should note that sine waves occur in the
daily life and we can relate many things to them.