Trans Sin Curve

7/23/2019 Trans Sin Curve

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Transformations of the Sine Curve

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

!otice that every aspect of the curve of y = sinx remained unchanged e"cept for the "-

values. The first point is #-30,0$ and !%T #0.0$. &s a result, each of the "-values of the

critical points followed suit and are all 30

0

to the left of the original value.

The table of values for y = sinx is

X 00 900 1800 2700 300

! 0 1 0 "1 0

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

X "300 00 1#00 2$00 3300

! 0 1 0 "1 0

&nother transformation that can be applied to y = sinx is a vertical translation. & verticaltranslation of -' will move the sinusoidal a"is to y ( -' 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 ne"t graph



!otice how the entire graph has moved downward. The blue line, which is locatedmidway between the ma"imum value of -) and the minimum value of -3, is the

sinusoidal a"is and its equation is y = "2% The only difference between the table of valuesfor 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 -', thus causing the ma"imum value to be -) andthe minimum value to be -3.


X 00 900 1800 2700 300

! 0 1 0 "1 0

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

X 00 900 1800 2700 300

! "2 "1 "2 "3 "2

&ll of the graphs that have been plotted thus far have an amplitude of one. *y applying a

vertical stretch to the graph of y = sinx' the amplitude can be increased or decreased.

& vertical stretch is shown in the equation as '

)

y = sinx. This will produce a graph thatwill have an amplitude of '. This means that the distance from the sinusoidal a"is to

either the ma"imum value or the minimum value will be '. & vertical stretch of'

) is

shown in the equation as 2y = sinx. This will produce a graph that has an amplitude of

'

) and the distance from the sinusoidal a"is to either the ma"imum value or the



minimum value will be'

). The graphs below will show these transformations

respectively

+hen the table of values of y = sinx is compared to the table of values of'

)y = sinx, the

only difference is the y-values of the ma"imum and minimum points are now ' and -'

respectively.


X 00 900 1800 2700 300

! 0 1 0 "1 0

The table of values for'

)y = sinx is

X 00 900 1800 2700 300

! 0 2 0 "2 0



+hen the table of values of y = sinx is compared to the table of values of 2y = sinx, the

only difference is the y-values of the ma"imum and minimum points are now'

) and

'

)− respectively.


X 00 900 1800 2700 300

! 0 1 0 "1 0

The table of values for 2y = sinx is

X 00 900 1800 2700 300

! 0 1

2

0 "1

2

0

The period or the domain over which one cycle of the curve is drawn is 300. owever,

the curve can undergo a horizontal stretch which will produce an increase or a decrease in

the period. & horizontal stretch of ' will produce one cycle of the graph of y = sinx thatis plotted over a domain of '00.

This transformation is written in the equation as y = sin '

)

x%

%n the other hand, a horizontal stretch of'

)will produce one cycle of the graph of

y = sinx that is plotted over a domain of )/00. This transformation is written in the

equation as y = sin2x. The ne"t graphs will show these transformations respectively



The stretch of the curve can be seen quite clearly. %ne cycle of the curve is drawn such

that 00.'00 ≤≤ X . Therefore the "-values in the table of values for y = sinx will all be

doubled in the table of values for y = sin '

)

x%

The table of values for y = sinx

X 00 900 1800 2700 300

! 0 1 0 "1 0

The table of values for y = sin'

)x is

X 00 1800 300 #$00 7200

! 0 1 0 "1 0



The stretch of the curve can be seen quite clearly. %ne cycle of the curve is drawn such

that 00 )/00 ≤≤ X . Therefore the "-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 300

! 0 1 0 "1 0

The table of values for y = sin2x is

X 00 $#0 900 13#0 1800

! 0 1 0 "1 0

&ll 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 thetransformations is written in a form called transformational form. Transformational form

of the equation y = sinx is written as ( ) ( )...

)sin..

..

)T H x

S H T V y

S V −=− . The

transformations of y = sinx can be listed from the equation. !ot all of the transformations

need to be applied to the function at once. There can be as few as one and as many asfive transformations applied to the graph of y = sinx%

"ample ) 1or the following function, list the transformations of y ( sin" and use thesetransformations to draw the graph.

$30sin#$3#') 0−=− x y

NO RV =.. '.. =S V 3.. =T V ).. =S H 30.. +=T H

There is no negative sign in front of $3#'

)− y so there is no vertical reflection. There is

a vertical stretch of ' which indicates that the amplitude of the curve will be two from the

sinusoidal a"is of y ( 3. The horizontal stretch of ) means that the graph will e"tend over

300 with the first "-value being 2300. &s a result of these transformations, the "-a"is

will begin at 00 and continue to 300. The y-a"is will begin at 0 and e"tend to 24. The

sinusoidal a"is will be located at y ( 3. The first point of #300, 3$ will be plotted and the



remaining 5 points will be plotted at 00intervals on the line y ( 3. The second point will

be moved upward to become #)'00, 4$ and the fourth point will be moved downward to

become #3000, )$. 1ollowing these steps will produce the following graph


X 00 900 1800 2700 300

! 0 1 0 "1 0

The table of values for $30sin#$3#'

) 0−=− x y is

X 300 1200 2100 3000 3900

! 3 # 3 1 3

"ample ' 1or the following function, list the transformations of y ( sin" and use these

transformations to draw the graph.

$50sin#$5# 0+=+− x y

050..

)..

5..

)..

..

−=

=

−=

=

=

T H

S H

T V

S V

YES RV

There is a negative sign in front of the e"pression #y 2 5$ which indicates that there is a

vertical reflection. The curve will have an amplitude of ) from the sinusoidal a"is of

y ( -5. The curve will be drawn over 300 with the first "-value located at -500. The "-

a"is will begin at -500 and e"tend to 3'00. The y-a"is will e"tend from -4 to 24. The

sinusoidal a"is will be the line y ( -5. The first point of #-500, -5$ will be plotted and the



remaining 5 points will be plotted at 00intervals on the line y ( -5. The second point

will be moved downward ) to become #400, -4$ and the fourth point will be moved

upward ) to become #'300, -3$. 1ollowing these steps will result in this graph


X 00 900 1800 2700 300

! 0 1 0 "1 0

The table of values for $50sin#$5# 0+=+− x y is

X "$00 #00 1$00 2300 3200

! "$ "# "$ "3 "$

"ample 3 1or the following function, list the transformations of y ( sin" and use these

transformations to draw the graph.

$0#'

)sin$#

3

) 0+=+ x y

The transformations of y = sinx are

These transformations can now be used to draw the graph. 1irst draw the a"es required

for the graph by considering both the horizontal and the vertical stretches. The "-a"is

0,0..

'..

,..

3..

..

−=

=

−=

=

=

T H

S H

T V S V

NO RV There is no negative sign in front of the e"pression $#

3

)+ y so there is !% vertical

reflection. The vertical stretch of 3 means that the amplitude of the curve is 3. Thevertical translation of - means that the sinusoidal a"is is located down si" and its

equation is y ( -. The horizontal stretch of ' means that the curve will be graphedover '00. The horizontal stretch is a factor that multiplies one period of the curve. The

horizontal translation of -00 means that the "-value of the first point #the phase shift$ is

-00.



must e"tend to '00 since the horizontal stretch is ' but it must begin at -00 to

accommodate the horizontal translation. The y-a"is must e"tend from 23 to -3 since the

vertical stretch is 3 but this stretch must be applied to the sinusoidal a"is of y ( -.

Therefore the y-a"is must e"tend from -3 to -. owever, to show the "-a"is, the y-a"is

must be drawn from 0 to -. %nce the a"es have been completed, the line y ( - should be

graphed. This can be done using a bro6en line since its presence is not necessary for the

graph. !ow the point #-00,-$ can be plotted. 1our other points are needed to complete

the graph which must be drawn over '00. To determine the interval between the points,

divide '00 by 5. *eginning at -00, add )/00 and plot the second point on the line

y ( -. 7ontinue this until five points have been plotted. The second and fourth points of

the graph produce the ma"imum and the minimum values of the curve. 8ove the second

point up 3 units from y ( - and the fourth point down 3 units from y ( -. There are now

five points plotted to form the graph of $0#'

)sin$#

3

) 0+=+ x y .


X 00 900 1800 2700 300

! 0 1 0 "1 0

The table of values for $0#'

)sin$#

3

) 0+=+ x y is



X "00 1200 3000 $800 00

! " "3 " "9 "

"ample 5 1or the equation $'0#'sin$#3 0

−=−− x y , list the transformations of

y = sinx and use these transformations to draw the graph.

0'0..

'

)..

..

3

)..

..

+=

=

+=

=

=

T H

S H

T V

S V

YES RV

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 3

)

. The graph will

move up from the "-a"is to the line y ( . The entire graph will be drawn over one-half

of 300 with the first "-value being located at '00. If )/00 is divided by 5, the interval

between the five points is 540. The remaining four points can be plotted on the line y ( .

The second point must be placed3

) downward from y ( and the fourth point must be

placed3

) upward from the sinusoidal a"is. The points can now be 9oined to form the

smooth curve that represents the graph of $'0#'sin$#3 0

−=−− x y .




X 00 900 1800 2700 300

! 0 1 0 "1 0

The table of values for $'0#'sin$#3 0

−=−− x y is

X 200 #0 1100 1##0 2000

! 7 %7 7 7%33 7

"ample 4 1or the following function, list the transformations of y ( sin" and use these

transformations to draw the graph

( ) ( )0)0'sin)3

) +=−− x y

0)0..

'

)..

)..

3..

..

−=

=

=

=

=

T H

S H

T V

S V

YES RV

**o the stu,ents to -rovi,e the ste-s ne.essary to .onstru.t the fo**oin/ /ra-h

This i** /ive them an o--ortunity to exhiit their un,erstan,in/ of set.hin/ the

/ra-h y usin/ the transformations of y = sinx


X 00 900 1800 2700 300

! 0 1 0 "1 0



The table of values for ( ) ( )0)0'sin)3

) +=−− x y is

X "100 3#0 800 12#0 1700

! 1 "2 1 $ 1

&ll of the graphs presented thus far, have been shown with a white bac6ground. owever,

when students create a graph using pencil and paper, it is best if they use grid paper. This

simplifies the plotting of the transformations. :rid paper is also used when students are

presented with a graph and as6ed to list the transformations or to write the equation that

models the graph. +hen analyzing a graph, the grid paper allows the students to see the

e"act numbers ; thus eliminating estimation the values.

"ample ) 1or the following graph we will list the transformations of y = sinx and

e"plain how each was determined.

The graph is not a reflection. The graph curves upward first and then downward. The

graph has a sinusoidal a"is of y ( 4 and the distance from the sinusoidal a"is to the

ma"imum point is '. Therefore, it has a vertical stretch of '. The graph begins at )40 and

ends at )340 producing a period of )340- )40 ( )'00.

This ma6es the horizontal stretch )'00 or ). The first "-value or phase shift of the

300 3

graph is )40 which means the horizontal translation is )40.



Transformations of y = sinx

0)4..

3)..

4..

'..

..

=

=

=

=

=

T H

S H

T V

S V

NO RV

"ample ' 1or this ne"t graph, we will simply list the transformations of y = sinx

because the students should now understand how to determine each change from the

graph.

!otice that the sinusoidal a"is was not drawn in this graph. The students must learn thatthe location of the a"is is midway between the ma"imum and minimum points.

The transformations of y = sinx

"ample 3 1or the following graph, list the transformations of y = sinx%

00..

3

)..

,..

5..

..

−=

=

−=

=

=

T H

S H

T V

S V

YES RV




"ample 5 1or the following graph, list the transformations of y = sinx%


030..

'

)..

5..

'

)..

..

−=

=

=

=

=

T H

S H

T V

S V

NO RV

0)0..

5

)..

3..

)..

..

=

=

−=

=

=

T H

S H

T V

S V

YES RV



xer.ises

). 1or each of the following equations, list the transformations of y = sinx and s6etch the

graph.

a$ x y 'sin' =− b$ ( ) ( )050'

)sin'

3

) +=−− x y

c$ ( ) ( )030sin4' −=+ x y d$ ( ) ( )0)43sin3'

) −=+ x y

'. 1or each of the following graphs, list the transformations of y = sinx%

a$

b$

c$



d$

So*utions



).a$


b$


50..

'..

'..

3..

..

−=

=

=

=

=

T H

S H

T V

S V

YES RV

NONE T H

S H

NONE T V

S V

YES RV

=

=

=

=

=

..

'

)..

..

'

)..

..



c$


d$


30..

)..

4..

'

)..

..

=

=

−=

=

=

T H

S H

T V

S V

NO RV

)4..

3

)..

3..

'..

..

=

=

−=

=

=

T H

S H

T V

S V

NO RV



'.

a$ The transformations of y = sinx

b$ The transformations of y = sinx

c$ The transformations of y = sinx

d$ The transformations of y = sinx

Thus far, all the transformations of y = sinx have been determined from the graph.

&nother way to determine the transformations is from the equation. <ecall that the

)4..

)..

3..

5..

..

−=

=

=

=

=

T H

S H

T V

S V

NO RV

'0..

'

)..

5..

'

)..

..

=

=

−=

=

=

T H

S H

T V

S V

YES RV

)0..

5

)..

..

3..

..

=

=

=

=

=

T H

S H

NONE T V

S V

YES RV

30..

3

)..

)..

4..

..

−=

=

−=

=

=

T H

S H

T V

S V

NO RV



transformational form of y = sinx is ( ) ( )...

)sin..

..

)T H x

S H T V y

S V −=− If numbers

are put into this equation, the transformations can be listed. owever, when the

transformations are listed there are mathematical concepts that must be applied.

ere is an e"ample that will show those concepts.

( ) ( )0305sin.'

) +=− x y

The transformations of y = sinx NO RV =.. There is not a ne/ative si/n in front of

'

)

'.. =S V This is the ,enominator of..

)

S V an, the

re.i-ro.a* of'

)

.. =T V The ne/ative si/n i** on*y remain

ne/ative

if ..T V is a -ositive va*ue% The ..T V is

a.tua**y the o--osite of hat is ritten

insi,e the ra.et%

5

).. =S H The $ e.omes the ,enominator of

..

)

S H %

This is the re.i-ro.a* of 5.

030.. −=T H The ne/ative si/n .han/es to a -ositive

if

the va*ue of ..T H is ne/ative% The..T H is

a.tua**y the o--osite of hat is ritten

insi,e the ra.et%

The numbers represent the transformations of y = sinx% If there is no number in front of

( )..T V y − or ( )..T H x − , then these transformations should be listed as %! and not

zero.

& rule to follow is list the ....= S H and S V sS → as reciprocals of what appears in the

equation and the ....= T H and T V sT → 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. &s you move the listed transformations from a list and into an equation, enter them

as reciprocals and opposites.

"ample ) 1or each of the following equations, list the transformations of y = sinx%

a$ ( ) ( )054'sin3

) −=+− x y b$ ( ) ( )00

3

)sin54 +=− x y

YES RV =.. 3.. =S V .. −=T V NO RV =.. 4

).. =S V

5.. =T V

'

).. =S H 054.. =S H 3.. =S H 00.. −=T H

c$ ( ) ( )0)05sin' −=−− x y d$ ( ) ( )0'0sin43 −=+ x y

YES RV =.. ).. =S V '.. =T V NO RV =.. 3

).. =S V

4.. −=T V

5

).. =S H 0

)0.. =T H ).. =S H 0'0.. =T H

>ust 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 theequation.

"ample ' 1or each of the following lists of transformations of y = sinx' write an

equation in transformational form to model each.

a$

030..

'

)..

4..

3..

..

=

=

=

=

=

T H

S H

T V

S V

NO RV

$30#'sin$4#3

) 0−=− x y

b$



0

'4..

'..

3..

5

)..

..

−=

=

−=

=

=

T H

S H

T V

S V

YES RV

$'4#'

)sin$3#5 0+=+− x y

c$

NONE T H

S H

T V

S V

NO RV

=

=

−=

=

=

..

3

)..

5..

)..

..

x y 3sin$5# =+

d$

0,0..

5..

,..

5..

..

=

=

=

=

=

T H

S H

T V

S V

YES RV

$0#5

)sin$#

5

) 0−=−− x y

The graph of y = sinx, the transformations of y = sinx, and the equation in

transformational form have all been addressed. %ne more area that must be e"amined is

the ma--in/ ru*e. & mapping rule e"plains e"actly 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 s6etch the graph. The

same concepts of reciprocals and opposites apply when writing a mapping rule to

represent the equation.

"ample 3 1or each of the following equations, write a mapping rule and create a table

of values

a) $)0sin#$5#'

) 0+=−− x y ) $'0#'

)sin$/#' 0−=+ x y .) $30#3sin$4#

0+=−− x y

( ) ( )5',)0, +−−→ y x y x ( )

−+→ /

'

),'0', y x y x ( )

+−−→ 4,303

), y x y x



,) $)/0#5

)sin

3

) 0+= x y

( ) ( ) y x y x 3,)/05, −→

These ma--in/ ru*es .an no e use, to .reate a ta*e of va*ues% ist the ori/ina*

va*ues of x an, y an, then a--*y the ma--in/ ru*e to /enerate the ne va*ues%

a) ( ) ( )5',)0, +−−→ y x y x

x→ (X"

10)

!→ ("2y+$)

00→ "10 0→ $

900

→ 80 1→ 21800

→

170 0→ $

2700

→

20 "1→

300

→

3#0 0→ $

)( )

−+→ /

'

),'0', y x y x

x→

(2X+20)

!→ (%#y"8)

00→ 20 0→ "8%0

900→ 200 1→ "7%#

1800

→

380 0→ "8%0

2700

→

#0 "1→ "8%#

300

→

7$0 0→ "8%0

.) ( )

+−−→ 4,303

), y x y x

x→ (1x"30) !→ ("y+#)



3

00→ "30 0→ #

900→ 0 1→ $

1800

→

30 0→ #

2700

→0 "1→

300

→

90 0→ #

,) ( ) ( ) y x y x 3,)/05, −→

x→ ($X"

180)

!→ (3y)

00→ "180 0→ 0900→ 180 1→ 3

1800

→

#$0 0→ 0

2700

→

900 "1→ "3

300

→

120 0→ 0

The points that are in blue are the points that would be plotted to s6etch the graph. These

points represent the five critical values for the graph of y = sinx that have undergone

transformations. ?tudents who have difficulty s6etching the graph by applying the

transformations to y = sinx use this method to create the graph. :raphing 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.

Trans Sin Curve

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