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# trigonometric functions 11 by 0Ccf04c

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```									                                    Trigonometric Functions
The graph of a trigonometric function such as y = sinx or y = cosx is a smooth curve. The graph
of y = sinx where x is an angle such that 0 0  x  3600 and y is the sine of each angle, can be
drawn by plotting the following points:

1.5   y

1

0.5
x
90                   180                 270                  360

–0.5

–1

–1.5

The graph of y = sinx can also be reflected in the x-axis. This will create a graph in which the
above y-values take on opposite signs. The equation would be written as
-y = sinx. and the table of values would be as follows:

The reflection of y = sinx is shown below:

1.5   y

1

0.5
x
30   60     90     120     150    180   210    240    270    300    330    360

–0.5

–1

–1.5
Using the points from the table, allow the students to see the formation of the curve as the points
are connected. These points represent one cycle of the function. Once the students are familiar
with the shape of the curve, they can reduce the number of points to the five critical values that
are used to draw the graph of y = sinx. These five critical values or points are:

X            00              900              1800           2700            3600
Y            0                1                0              -1              0

Students should now examine the graph to note its characteristics. The graph has a maximum
value of 1 at x = 900 and a minimum value of -1 at x = 2700. The graph also has three points
plotted on the line y = 0 at x = 00 , x = 1800 and at x = 3600.
This line, y = 0, is located midway between the maximum and minimum values.
The name given to this line is the sinusoidal axis.

The distance from the sinusoidal axis to either the maximum value or to the minimum value is
called the amplitude of the curve.

To graph one cycle of the sine curve requires 3600. This represents the period of the sine curve.

The first point that was plotted was (00, 0). If the beginning point moves to the left or right of 00,
the amount of movement is known as the phase shift.
The graph of y = sinx will repeat every 3600 and is therefore periodic. This can be seen in the
graph below as one cycle appears from 00 to -3600 and another from 00 to 3600.

1.5   y

1

0.5
x
–360       –270       –180        –90                   90        180        270        360

–0.5

–1

–1.5

The graph of y = sinx can undergo transformations that will directly affect the characteristics of
the graph. If the graph is moved 300 to the left, it undergoes a horizontal translation that will
produce a phase shift of -300. This transformation is shown in the equation as y = sin(x + 300)
and in the graph below:

Notice that every aspect of the curve of y = sinx remained unchanged except for the x-values.
The first point is (-30,0) and NOT (0.0). As a result, each of the x-values of the critical points
followed suit and are all 300 to the left of the original value.

The table of values for y = sinx is:

X               00                900          1800            2700            3600
Y               0                  1            0               -1              0

The table of values for y = sin(x + 300) is:

X              -300               600          1500            2400            3300
Y                0                 1            0               -1              0

Another transformation that can be applied to y = sinx is a vertical translation. A vertical
translation of -2 will move the sinusoidal axis to y = -2 and in turn the entire graph will be
moved downward two units. This change is shown in the equation as (y + 2) = sinx.
This transformation is shown in the next graph:
y
1

x
30   60        90   120    150   180    210    240    270      300   330      360

–1

–2

–3

Notice how the entire graph has moved downward. The blue line, which is located midway
between the maximum value of -1 and the minimum value of -3, is the sinusoidal axis and its
equation is y = -2. The only difference between the table of values for the graph of y = sinx and
the tale of values for the graph of (y + 2) = sinx is that the y-values of 0 in the initial table are
now -2, thus causing the maximum value to be -1 and the minimum value to be -3.

The table of values for y = sinx is:

X             00              900          1800             2700               3600
Y             0                1            0                -1                 0

The table of values for (y + 2) = sinx is:

X             00              900          1800             2700               3600
Y             -2               -1           -2               -3                 -2

All of the graphs that have been plotted thus far have an amplitude of one. By applying a
vertical stretch to the graph of y = sinx, the amplitude can be increased or decreased.
1
A vertical stretch is shown in the equation as  y = sinx. This will produce a graph that will
2
have an amplitude of 2. This means that the distance from the sinusoidal axis to either the
1
maximum value or the minimum value will be 2. A vertical stretch of       is shown in the
2
1
equation as 2y = sinx. This will produce a graph that has an amplitude of     and the distance
2
1
from the sinusoidal axis to either the maximum value or the minimum value will be         . The
2
graphs below will show these transformations respectively:

y
2

1

x
30   60        90    120    150   180   210    240   270     300   330      360

–1

–2

1
y = sinx, the only
When the table of values of y = sinx is compared to the table of values of
2
difference is the y-values of the maximum and minimum points are now 2 and -2

respectively.

The table of values for y = sinx is:

X             00               900          1800          2700               3600
Y             0                 1            0             -1                 0

1
The table of values for       y = sinx is:
2

X             00               900          1800          2700               3600
Y             0                 2            0             -2                 0
1    y

0.75

0.5

0.25
x
30    60        90   120    150   180   210    240    270     300   330      360
–0.25

–0.5

–0.75

–1

When the table of values of y = sinx is compared to the table of values of 2y = sinx, the only
1       1
difference is the y-values of the maximum and minimum points are now         and      respectively.
2        2

The table of values for y = sinx is:

X              00              900          1800           2700               3600
Y              0                1            0              -1                 0

The table of values for 2y = sinx is:

X              00              900          1800           2700               3600
Y              0                1            0              -1                 0
2                           2

The period or the domain over which one cycle of the curve is drawn is 3600. However, the
curve can undergo a horizontal stretch which will produce an increase or a decrease in the
period. A horizontal stretch of 2 will produce one cycle of the graph of y = sinx that is plotted
over a domain of 7200.
1
This transformation is written in the equation as y = sin x.
2
1
On the other hand, a horizontal stretch of will produce one cycle of the graph of           y=
2
sinx that is plotted over a domain of 1800. This transformation is written in the
equation as y = sin2x. The next graphs will show these transformations respectively:
1.5   y

1

0.5
x
60    120        180   240    300   360   420    480   540     600   660      720

–0.5

–1

–1.5

The stretch of the curve can be seen quite clearly. One cycle of the curve is drawn such that
0 0  X  7200 . Therefore the x-values in the table of values for y = sinx will all be doubled in
1
the table of values for y = sin x.
2

The table of values for y = sinx:

X               00               900          1800          2700               3600
Y               0                 1            0             -1                 0

1
The table of values for y = sin x is:
2

X               00               1800         3600          5400               7200
Y               0                 1            0             -1                 0

1.5   y

1

0.5
x
30               60           90           120           150            180

–0.5

–1

–1.5
The stretch of the curve can be seen quite clearly. One cycle of the curve is drawn such that
0 0  X  1800 . Therefore the x-values in the table of values for y = sinx will all be multiplied
by one-half in the table of values for y = sin2x.

The table of values for y = sinx:

X               00                900               1800           2700             3600
Y               0                  1                 0              -1               0

The table of values for y = sin2x is:

X               00                450               900            1350             1800
Y               0                  1                 0              -1               0

All of the transformations that can be applied to the graph of y = sinx have been shown
graphically and as they appear in the equation. The equation of y = sinx that shows the
transformations is written in a form called transformational form. Transformational form of the
equation y = sinx is written as
1
 y  V .T .  sin 1 x  H .T . . The transformations of y =
V .S .                    H .S
sinx can be listed from the equation. Not all of the transformations need to be applied to the
function at once. There can be as few as one and as many as five transformations applied to the
graph of y = sinx.

Example 1: For the following function, list the transformations of y = sinx and use these
transformations to draw the graph.

1
( y  3)  sin( x  30 0 )
2

V .R.  NO
V .S.  2
V .T .  3
H.S.  1
H .T .  30

1
There is no negative sign in front of       ( y  3) so there is no vertical reflection. There is a vertical
2
stretch of 2 which indicates that the amplitude of the curve will be two from the sinusoidal axis
of y = 3. The horizontal stretch of 1 means that the graph will extend over 3600 with the first x-
value being +300. As a result of these transformations, the x-axis will begin at 00 and continue to
3900. The y-axis will begin at 0 and extend to +5. The sinusoidal axis will be located at y = 3.
The first point of (300, 3) will be plotted and the remaining 4 points will be plotted at
900intervals on the line y = 3. The second point will be moved upward to become (1200, 5) and
the fourth point will be moved downward to become (3000, 1). Following these steps will
produce the following graph:

6   y

5

4

3

2

1
x
30   60    90     120     150     180    210     240   270   300    330   360   390

The table of values for y = sinx is:

X             00                900                1800          2700           3600
Y             0                  1                  0             -1             0

1
The table of values for      ( y  3)  sin( x  30 0 ) is:
2
X             300              1200                2100          3000           3900
Y              3                 5                  3             1              3

Example 2: For the following function, list the transformations of y = sinx and use these
transformations to draw the graph.

 ( y  4)  sin( x  40 0 )
V .R.  YES
V .S .  1
V .T .  4
H .S .  1
H .T .  40 0

There is a negative sign in front of the expression (y + 4) which indicates that there is a vertical
reflection. The curve will have an amplitude of 1 from the sinusoidal axis of            y = -4. The
curve will be drawn over 3600 with the first x-value located at -400. The x-axis will begin at -400
and extend to 3200. The y-axis will extend from -5 to +5. The sinusoidal axis will be the line y
= -4. The first point of (-400, -4) will be plotted and the remaining 4 points will be plotted at
900intervals on the line y = -4. The second point will be moved downward 1 to become (500, -5)
and the fourth point will be moved upward 1 to become (2300, -3). Following these steps will
result in this graph:

5   y

x
–30                   30          60      90     120     150      180   210   240    270   300

–5

The table of values for y = sinx is:

X                        00                 900               1800         2700         3600
Y                        0                   1                 0            -1           0

The table of values for  ( y  4)  sin( x  40 0 ) is:

X                       -400                500               1400         2300         3200
Y                        -4                  -5                -4           -3           -4

Example 3: For the following function, list the transformations of y = sinx and use these
transformations to draw the graph.

1               1
( y  6)  sin ( x  600 )
3               2

The transformations of y = sinx are:
1
There is no negative sign in front of the expression    ( y  6) so there is NO vertical
V .R.  NO                                                                      3
V .S .  3                reflection. The vertical stretch of 3 means that the amplitude of the curve is 3. The
V .T .  6               vertical translation of -6 means that the sinusoidal axis is located down six and its
equation is y = -6. The horizontal stretch of 2 means that the curve will be graphed
H .S .  2                over 7200. The horizontal stretch is a factor that multiplies one period of the curve.
H .T .  60 0            The horizontal translation of -600 means that the x-value of the first point (the phase
shift) is -600.
These transformations can now be used to draw the graph. First draw the axes required for the
graph by considering both the horizontal and the vertical stretches. The x-axis must extend to
7200 since the horizontal stretch is 2 but it must begin at -600 to accommodate the horizontal
translation. The y-axis must extend from +3 to -3 since the vertical stretch is 3 but this stretch
must be applied to the sinusoidal axis of y = -6. Therefore the y-axis must extend from -3 to -9.
However, to show the x-axis, the y-axis must be drawn from 0 to -9. Once the axes have been
completed, the line y = -6 should be graphed. This can be done using a broken line since its
presence is not necessary for the graph. Now the point (-600,-6) can be plotted. Four other points
are needed to complete the graph which must be drawn over 7200. To determine the interval
between the points, divide 7200 by 4. Beginning at -600, add 1800 and plot the second point on
the line        y = -6. Continue this until five points have been plotted. The second and fourth
points of the graph produce the maximum and the minimum values of the curve. Move the
second point up 3 units from y = -6 and the fourth point down 3 units from y = -6. There are now
1              1
five points plotted to form the graph of ( y  6)  sin ( x  600 ) .
3              2

y
x
–60            60    120    180   240       300   360     420   480   540    600   660   720

–2

–4

–6

–8

The table of values for y = sinx is:

X               00              900               1800          2700           3600
Y               0                1                 0             -1             0
1               1
The table of values for       ( y  6)  sin ( x  600 ) is:
3               2

X               -600              1200             3000             4800          6600
Y                -6                -3               -6               -9            -6

Example 4: For the equation  3( y  7)  sin 2( x  20 0 ) , list the transformations of
y = sinx and use these transformations to draw the graph.
V .R.  YES
1
V .S . 
3
V .T .  7
1
H .S . 
2
H .T .  20 0

There is a vertical reflection so the graph will go downward first and then curve upward. The

vertical stretch will produce a graph that has an amplitude of 1 . The graph will move up from
3
the x-axis to the line y = 7. The entire graph will be drawn over one-half of 3600 with the first x-
value being located at 200. If 1800 is divided by 4, the interval between the five points is 450.

The remaining four points can be plotted on the line y = 7. The second point must be placed 1
3

downward from y = 7 and the fourth point must be placed 1 upward from the sinusoidal axis.
3
The points can now be joined to form the smooth curve that represents the graph of
 3( y  7)  sin 2( x  20 0 ) .

10   y
9
8
7
6
5
4
3
2
1
x
20        40           60   80       100          120   140     160   180      200
The table of values for y = sinx is:

X                  00                   900                      1800               2700               3600
Y                  0                     1                        0                  -1                 0

The table of values for  3( y  7)  sin 2( x  20 0 ) is:

X                  200                 650                       1100               1550               2000
Y                   7                  6.67                       7                 7.33                7

Example 5: For the following function, list the transformations of y = sinx and use these
transformations to draw the graph


1
3

 y  1  sin 2 x  100     
V .R.  YES
V .S .  3
V .T .  1
1
H .S . 
2
H .T .  10 0

Allow the students to provide the steps necessary to construct the following graph: This will
give them an opportunity to exhibit their understanding of sketching the graph by using
the transformations of y = sinx

5   y

4

3

2

1
x
–10            10   20   30    40      50   60      70       80   90    100   110   120   130    140   150   160   170
–1

–2

–3
The table of values for y = sinx is:

X               00                900               1800       2700             3600
Y               0                  1                 0          -1               0

The table of values for 
1
3
       
 y  1  sin 2 x  100 is:

X              -100               350               800        1250             1700
Y                1                 -2                1          4                1

All of the graphs presented thus far, have been shown with a white background. However, when
students create a graph using pencil and paper, it is best if they use grid paper. This simplifies the
plotting of the transformations. Grid paper is also used when students are presented with a graph
and asked to list the transformations or to write the equation that models the graph. When
analyzing a graph, the grid paper allows the students to see the exact numbers – thus eliminating
estimation the values.

Example 1: For the following graph we will list the transformations of y = sinx and
explain how each was determined.

The graph is not a reflection. The graph curves upward first and then downward. The graph has a
sinusoidal axis of y = 5 and the distance from the sinusoidal axis to the maximum point is 2.
Therefore, it has a vertical stretch of 2. The graph begins at 150 and ends at 1350 producing a
period of 1350- 150 = 1200.
This makes the horizontal stretch 1200 or 1. The first x-value or phase shift of the
3600    3
graph is 150 which means the horizontal translation is 150.

V .R.  NO
V .S .  2
Transformations of y = sinx:               V .T .  5
1
H .S . 
3
H .T .  15 0

Example 2: For this next graph, we will simply list the transformations of y = sinx because the
students should now understand how to determine each change from the graph.

Notice that the sinusoidal axis was not drawn in this graph. The students must learn that the
location of the axis is midway between the maximum and minimum points.

The transformations of y = sinx:            V .R.  YES
V .S .  4
V .T .  6
1
H .S . 
3
H .T .  90 0
Example 3: For the following graph, list the transformations of y = sinx.

V .R.  NO
1
The transformations of y = sinx:         V .S . 
2
V .T .  4
1
H .S . 
2
H .T .  30 0

Example 4: For the following graph, list the transformations of y = sinx.
V .R.  YES
The transformations of y = sinx:             V .S .  1
V .T .  3
1
H .S . 
4
H .T .  10 0
Exercises:

1. For each of the following equations, list the transformations of y = sinx and sketch the
graph.

a)  2 y  sin 2 x                        b) 
1
3                2

 y  2  sin 1 x  400      

c) 2 y  5  sin x  30 0                  d)
1
2

 y  3  sin 3 x  150   
2. For each of the following graphs, list the transformations of y = sinx.

a)

b)
c)

d)
Solutions:

1.a)

The transformations of y = sinx:   V .R.  YES
1
V .S . 
2
V .T .  NONE
1
H .S . 
2
H .T .  NONE

b)

The transformations of y = sinx:
V .R.  YES
V .S .  3
V .T .  2
H .S .  2
H .T .  40
c)

The transformations of y = sinx:   V .R.  NO
1
V .S . 
2
V .T .  5
H .S .  1
H .T .  30
d)

The transformations of y = sinx:    V .R.  NO
V .S .  2
V .T .  3
1
H .S . 
3
H .T .  15
2.
V .R.  NO
V .S .  4
a) The transformations of y = sinx:
V .T .  3
H .S .  1
H .T .  15

V .R.  YES
1
b) The transformations of y = sinx:   V .S . 
2
V .T .  4
1
H .S . 
2
H .T .  20

V .R.  NO
V .S .  5
V .T .  1
c) The transformations of y = sinx:
1
H .S . 
3
H .T .  30

V .R.  YES
V .S .  3
d) The transformations of y = sinx:   V .T .  NONE
1
H .S . 
4
H .T .  10
Thus far, all the transformations of y = sinx have been determined from the graph. Another way
to determine the transformations is from the equation. Recall that the transformational form of y

= sinx is
1
 y  V .T .  sin 1 x  H .T . If numbers are put into this equation, the
V .S .                    H .S
transformations can be listed. However, when the transformations are listed there are
mathematical concepts that must be applied.

Here is an example that will show those concepts.

1
2

 y  7  sin 4 x  300   
1
The transformations of y = sinx: V .R.  NO There is not a negative sign in front of
2

1
V .S.  2 This is the denominator of             and the
V .S .
1
reciprocal of
2

V .T .  7 The negative sign will only remain negative
if V .T . is a positive value. The V .T . is
actually the opposite of what is written
inside the bracket.

1                                   1
H .S .      The 4 becomes the denominator of        .
4                                  H .S .
This is the reciprocal of 4.

H .T .  300 The negative sign changes to a positive if
the value of H .T . is negative. The H .T . is
actually the opposite of what is written
inside the bracket.

The numbers represent the transformations of y = sinx. If there is no number in front of
 y  V .T . or x  H .T . , then these transformations should be listed as ONE and not zero.
A rule to follow is list the S ' s  V .S.and H .S. as reciprocals of what appears in the equation
and the T ' s  V .T .and H .T . as the opposite of what appears in the equation.
This same rule can be used to write an equation of a sinusoidal curve in transformational form.
As you move the listed transformations from a list and into an equation, enter them as reciprocals
and opposites.

Example 1: For each of the following equations, list the transformations of y = sinx.
1
a)
3

 y  6  sin 2 x  450                                         
b) 5 y  4  sin x  60 0
1
3

1
V .R.  YES V .S.  3 V .T .  6                  V .R.  NO V .S .       V .T .  4
5
1
H .S .         H .S.  450                         H .S.  3 H .T .  600
2

c)                       
  y  2  sin 4 x  10 0                  d) 3 y  5  sin x  20 0 

1
V .R.  YES V .S.  1 V .T .  2                       V .R.  NO V .S .        V .T .  5
3
1
H .S .         H .T .  100                             H.S.  1 H .T .  200
4

Just as the transformations were listed using the reciprocals and the opposites coming from the
equation to the list, the same process is followed going from the list to the equation.

Example 2: For each of the following lists of transformations of y = sinx, write an
equation in transformational form to model each.

a)
V .R.  NO
V .S .  3
1
V .T .  5                            ( y  5)  sin 2( x  30 0 )
3
1
H .S . 
2
H .T .  30 0
b)
V .R.  YES
1
V .S . 
4
1
V .T .  3                         4( y  3)  sin ( x  250 )
2
H .S .  2
H .T .  25 0

c)

V .R.  NO
V .S .  1
V .T .  4                          ( y  4)  sin 3x
1
H .S . 
3
H .T .  NONE

V .R.  YES
V .S .  4
1              1
d) V .T .  6                           ( y  6)  sin ( x  600 )
4              4
H .S .  4
H .T .  60 0

The graph of y = sinx, the transformations of y = sinx, and the equation in transformational form
have all been addressed. One more area that must be examined is the mapping rule. A mapping
rule explains exactly what happened to the original points of y = sinx when transformations
occurred. The mapping rule can be used to create a table of values for the equation. These points
can then be plotted to sketch the graph. The same concepts of reciprocals and opposites apply
when writing a mapping rule to represent the equation.
Example 3: For each of the following equations, write a mapping rule and create a table
of values:

1                                             1
a)  ( y  4)  sin( x  10 0 ) b) 2( y  8)  sin ( x  20 0 )       c)  ( y  5)  sin 3( x  30 0 )
2                                             2

x, y   x  10,2 y  4        x, y    2 x  20 , 1 y  8 
                         x, y    1 x  30 , y  5 
                   
           2                       3                      

1        1
d)     y  sin ( x  1800 )
3        4

x, y   4 x  180 ,3 y 

These mapping rules can now be used to create a table of values. List the original values of
x and y and then apply the mapping rule to generate the new values.

a) x, y   x  10 ,2 y  4
x       (X-10) Y  (-2y+4)
0 
0
-10      0          4
90 0
     80      1          2
1800      170      0          4
270 
0
260     -1          6
360 
0
350      0          4

b)     x, y    2 x  20 , 1 y  8 
                    
      2   
x       (2X+20) Y  (.5y-8)
00        20    0    -8.0
900       200    1    -7.5
1800       380    0    -8.0
2700       560   -1    -8.5
3600       740    0    -8.0
c)     x, y    1 x  30 , y  5 
                   
3        
x        (1x-30) Y  (-y+5)
3
00        -30    0    5
900         0     1    4
1800        30     0    5
2700        60    -1    6
3600        90     0    5

d)     x, y   4 x  180 ,3 y 
x  (4X-180) Y                    (3y)
00      -180       0            0
90 
0
180       1            3
180 
0
540       0            0
2700        900      -1            -3
3600  1260            0            0

The points that are in blue are the points that would be plotted to sketch the graph. These points
represent the five critical values for the graph of y = sinx that have undergone transformations.
Students who have difficulty sketching the graph by applying the transformations to y = sinx use
this method to create the graph. Graphing was initially introduced to students as plotting points
and many are still more proficient at this method than any other. The end result is the same – the
graph of the sinusoidal curve.

The next sinusoidal curve to explore is that of y = cosx.

The table of values for y = cosx is:

X                 00                900        1800            2700            3600
Y                 1                  0          -1              0               1

Notice that the y-values of the five critical points are different than those of y = sinx.
The graph of y = cosx:

The parts of the curve, the transformations of y = cosx, writing a mapping rule and creating a
table of values are all done in the same format as those of y = sinx.

Example 4:

For each of the following equations, write a mapping rule, create a table of values and sketch the
graph.
Note: These are the same equations used in Example 3 with the function being the only
change.

1                                           1
a)  ( y  4)  cos(x  100 ) b) 2( y  8)  cos ( x  20 0 )      c)  ( y  5)  cos 3( x  30 0 )
2                                           2

x, y   x  10,  2 y  4   x, y    2 x  20 , 1 y  8 
                         x, y    1 x  30 ,  y  5 
                    
           2                        3                     

1        1
d)     y  cos ( x  1800 )
3        4

x, y   4 x  180 , 3 y 

These mapping rules can now be used to create a table of values. List the original values of
x and y and then apply the mapping rule to generate the new values. The x-values are the
same for both y = sinx and y = cosx. Notice that the y-values are different.
a) x, y   x  10 ,2 y  4
x       (X-10) Y  (-2y+4)
0 
0
-10      1          2
900       80      0          4
180 
0
170     -1          6
270 
0
260      0          4
3600      350      1          2

b)     x, y    2 x  20 , 1 y  8 
                    
      2   
x       (2X+20) Y  (.5y-8)
00        20    1    -7.5
900       200    0    -8.0
1800       380   -1    -8.5
2700       560    0    -8.0
3600       740    1    -7.5

c)     x, y    1 x  30 , y  5 
                   
3        
x        (1x-30) Y  (-y+5)
3
00        -30    1    4
900         0     0    5
1800        30    -1    6
2700        60     0    5
3600        90     1    4

d)     x, y   4 x  180 ,3 y 
x  (4X-180) Y                    (3y)
0 
0
-180       1            3
90 0
      180       0            0
180 
0
540      -1            -3
270 
0
900       0            0
360 0
 1260           1            3

1
a)  ( y  4)  cos(x  100 )
2
1
b) 2( y  8)  cos ( x  20 0 )
2
c)       ( y  5)  cos 3( x  30 0 )

1        1
d)     y  cos ( x  1800 )
3        4

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