webtrig flash cards - the university of akron
TRANSCRIPT
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Begin FS
Close Home
THE UNIVERSITY OF AKRONTheoretical and Applied Mathematics
Flash CardsGraphs of the sine and cosine
functionsKatie Jones
andTom Price
Instructions: Click on the Begin button to view thefirst randomly selected card. Click on FS to view thecards in full screen mode (works only outside a webbrowser). The Home button on the first page goes tothe WebTrig home page; otherwise, the Home buttonreturns to this page. The Close button closes the doc-ument (use outside a web browser).
c© 2003 [email protected] Revision Date: April 19, 2003 Version 1.0
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
State the amplitude of the func-tion
y (t) = −4 sin[π
3(t + π)
].
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
State the amplitude of the func-tion
y (t) =15
cos[3(t +
π
5
)].
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
State the period of the function
y (t) = 3 cos (2t + 6) + 7.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
State the period of the function
y (t) =23
sin(
−13t
)+
43.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
For the following function findthe phase shift and its direction
y (t) = − sin (t + 3π) + 1.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
For the following function findthe phase shift and its direction
y (t) = cos(
2π3
t − π
).
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Graph the function
y (t) = 3 sin t.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Graph the function
y (t) = − sin (2t) +12.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Graph the function
y (t) =14
sin(π
2t +
π
3
).
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Graph the function
y (t) = sin(t − π
4
)− 2.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Graph the function
y (t) = 4 cos(
−12t
).
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Graph the function
y (t) = −34
cos (2t + 4) .
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Graph the function
y (t) = cos (6t) − 3.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Graph the function
y (t) =14
cos[π
4(t − 1)
].
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Find a sine function whose graphlooks like the following.
�
�−
−1�
�
1
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Find a sine function whose graphlooks like the following
�
�� �
� � �
� �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Find a cosine function whosegraph is the following.
� �
� � � �
�
� � �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Find a cosine function that graphs�
� � �
�−
�−
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Find a cosine function whosegraph is identical to the graph ofthe function
y (t) = 5 sin(
12t +
π
2
).
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Find a sine function whose graphis identical to the graph of thefunction
y (t) =12
cos [π (t + 1)] + 1.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find the amplitude of the function
y (t) = −4 sin[π
3(t + π)
]remember it is defined as the maximum ver-tical deviation of the function’s graph fromthe t-axis.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: 4Solution: The function y (t) = −4 sin
(π3 t + π
)is in the
form f (t) = A sin [a (t + b)] . By definition, the amplitude off (t) is |A| , so for y (t) , the amplitude is
| − 4| = 4.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find the amplitude of the function
y (t) =15
cos[3(t +
π
5
)]recall this is defined as the maximum verticaldeviation of the function’s graph from the t-axis.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer:15
Solution: The function, y (t) = 15 cos
(3t + π
5
)is in the
form f (t) = A cos [a (t + b)] . By definition, the amplitude off (t) is |A| , so for y (t) , the amplitude is∣∣∣∣15
∣∣∣∣ =15.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find the period of the function
y (t) = 3 cos (2t + 6) + 7
recall it is defined as the distance measuredon the t-axis that it takes for the graph tocomplete one oscillation.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: π
Solution: The period of f (t) = A cos [a (t + b)] + c is de-fined to be 2π
|a| . Writing
y (t) = 3 cos (2t + 6) + 7 = 3 cos [2 (t + 3)] + 7
we see that a = 2 so y has period2π
|2| = π.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find the period of the function
y (t) =23
sin(
−13t
)+
43
recall that the period of
f (t) = A sin [a (t + b)] + c
is 2π|a| .
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: 6πSolution: The function, y (t) = 2
3 sin(− 1
3 t)
+ 43 is in the
form f (t) = A sin [a (t + b)] + c. The period of f (t) is definedto be 2π
|a| , so for y (t) , the period is
2π∣∣− 13
∣∣ = 6π.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find the phase shift including its directionfor the function
y (t) = − sin (t + 3π) + 1
remember the pahse shift is defined as thedistance a graph is translated along the t-axis (right or left).
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: 3π units to the leftSolution: The function, y (t) = − sin (t + 3π) + 1 is in the
form f (t) = A sin [a (t + b)] + c. The phase shift for f (t) is b,so for y (t) , the phase shift is
3π.
This moves the graph to the left since the phase shift is posi-tive.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find the phase shift including its directionfor the function
y (t) = cos(
2π3
t − π
)
put the function into the form f (t) =A cos [a (t + b)] + c and recall that the phaseshift is determined by b.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: −32 or 3
2 units to the right
Solution: The function, y (t) = cos( 2π
3 t − π)
needs to bein the form f (t) = A cos [a (t + b)] + c. So write,
cos(
2π
3t − π
)= cos
[2π
3
(t − 3
2
)].
The phase shift for f (t) is defined to be b, so for y (t) , thephase shift is
−32.
This moves the graph to the right since the phase shift isnegative. �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To graph the function
y (t) = 3 sin t
notice that the 3 causes a change in thegraph’s amplitude.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
� � �
�
� � � �
�
� � � � �
� �
Solution: The graph of y = 3 sin t has the same generalappearance as the fundamental sine function except that theamplitude is 3 units. �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To graph the function
y (t) = − sin (2t) +12
notice that there are three changes to thefundamental sine graph, one being that thenegative sign causes a reflection over the t-axis.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
�
�
� � �
�
�
�
�
� � �� � �� �
� ��
�
Solution: The graph of y = − sin (2t) + 12 has the same
general appearance as the fundamental sine function exceptthat the period is 2π
|2| = π, the entire graph is shifted up 12
unit, and the graph is reflected over the line y = 12 . �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To graph the function
y (t) =14
sin(π
2t +
π
3
)put it in the form
f (t) = A sin [a (t + b)] + c.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
� �
�− �� � �−
� � � �
� �
Solution: The function, y (t) = 14 sin
(π2 t + π
3
)needs to be
in the form f (t) = A sin [a (t + b)] + c. So write,
y (t) =14
sin(π
2t +
π
3
)=
14
sin[π
2
(t +
23
)]
Then, y has a period of2π∣∣π2
∣∣ = 4 and an amplitude of 14 . Also,
y has a phase shift of23
units to the left. �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To graph the function
y (t) = sin(t − π
4
)− 2
notice there are two changes in the funda-mental sine graph.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
�
��
�
�
�
�
�
� � �
� � � �� �
� �
Solution: The graph of y = sin(t − π
4
) − 2 has the samegeneral appearance as the fundamental sine function exceptthat there is a phase shift of π
4 units to the right and a verticalshift of 2 units down. �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To graph the function
y (t) = 4 cos(
−12t
)
notice there is a change in the period and theamplitude of the fundamental cosine graph.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
�
�
�−
�
π−π 3π2π 4π−2π
Solution: The graph of y = 4 cos(− 1
2 t)
has the same gen-eral appearance as the fundamental cosine function except
that the period is2π∣∣− 1
2
∣∣ = 4π units and the amplitude is 4.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To graph the function
y (t) = −34
cos (2t + 4)
notice that it is the result of four changesin the fundamental cosine graph, and thefunction needs to be in the form f (t) =A cos [a (t + b)] + c.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
y
t
� � ��
� � ���
� � � �
Solution: The function, y (t) = − 34 cos (2t + 4) needs to be
in the form f (t) = A sin [a (t + b)] + c. So write,
y (t) = −34
cos (2t + 4) = −34
cos [2 (t + 2)] .
Then, y has a period of2π
|2| = π and an amplitude of 34 . Also,
y has a phase shift of 2 units to the left and should be reflectedaround the t-axis because of the negative sign preceding theamplitude. �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To graph the function
y (t) = cos (6t) − 3
there is a change in the period and the ver-tical shift of the fundamental cosine graph.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
y
t
��
��
�� ��
� � � �
Solution: The graph of y = cos (6t) − 3 has the same gen-eral appearance as the fundamental cosine function except
that the period is2π
|6| =π
3units and there is a vertical shift
down 3 units. �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To graph the function
y (t) =14
cos[π
4(t − 1)
]notice there is a change in the period, phaseshift, and amplitude of the fundamental co-sine graph.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
y
t� � ��� � �
�
Solution: The graph of y = 14 cos
[π4 (t − 1)
]has the same
general appearance as the fundamental cosine function except
that the period is2π∣∣π4
∣∣ = 8 units, there is a phase shift of 1
unit to the right, and the amplitude is 14 unit. �
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find a sine function that whose graph isgiven in the figure first determine the period.
�
�−
−1�
�
1
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: y (t) = 2 sin(
π2t + π
)Solution: Since the graph of the function is symmetric with
respect to the t-axis and deviates from this axis by 2 units, itsamplitude is 2. Next, the distance measured along the t-axisfrom one peak to the next suggests that the period is 4 units.This means that 2π
a = 4 so that a = π2 . Noticing differences
between the given graph and the fundamental sine function,there is a phase shift of 2 units to the left. Using the formf (t) = A sin [a (t + b)] + c this indicates that
y (t) = 2 sin[π
2(t + 2)
]= 2 sin
(π
2t + π
).
There are other acceptable answers for this question including
ya (t) = −2 sin[π
2(t + 4)
]and yb (t) = 2 sin
[π
2(t − 2)
].
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find a sine function for the graph
�
�� �
� � �
� �
note any changes from the fundamental sinegraph.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: y (t) = 12 sin
(12t
)+ 1
Solution: There is a vertical shift of 1 unit up since thegraph is symmetric with respect to the line y = 1. The max-imum deviation of 1
2 from this line of symmmetry gives theamplitude of 1
2 units. The distance measured along the t-axisfrom one peak to the next means the period is 4π units. Since2πa = 4π we have a = 2π
4π = 12 . Hence,
y (t) =12
sin(
12t
)+ 1.
There are other acceptable answers for this question including
ya (t) = −12
sin(
t
2− π
)+1 and yb (t) =
12
sin(
t
2− 2π
)+1.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find a cosine function for the graph
� �
� � � �
�
� � �
determine the length of one oscillation of thewave.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: y (t) = 14 cos (t + π)
Solution: Find a peak of the graph which is the maximumdeviation from the t-axis, this gives the amplitude of 1
4 units.Noticing differences between the given graph and the funda-mental cosine graph, there is a phase shift of π units to theleft. Using the form f (t) = A cos [a (t + b)] + c this indicatesthat
y (t) =14
cos (t + π) .
There are other acceptable answers for this question including
ya (t) =14
cos (t − π) and yb (t) = −14
cos t.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
�
� � �
�−
�−
To find a cosine func-tion for the given graphnote any changes fromthe fundamental cosinegraph in the period,phase shift, amplitude,or vertical shift.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: y (t) = 3 cos (2πt)Solution: Find a peak of the graph which is the maximum
deviation from the t-axis, this gives the amplitude of 3 units.Next, the distance measured along the t-axis from one peak tothe next is the period of the function. In this case the periodis 1 unit. This means that 2π
a = 1 so that a = 2π.Using theform f (t) = A cos [a (t + b)] + c this indicates that
y (t) = 3 cos (2πt) .
There are other acceptable answers for this question including
ya (t) = −3 cos[2π
(t +
12
)]and yb (t) = 3 cos [2π (t + 1)] .
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find a cosine function whose graph is iden-tical to the graph of the function y (t) =5 sin
(12t + π
2
)use an appropriate phase shift.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: y (t) = 5 cos(1
2t)
Solution: The function, y (t) = 5 sin( 1
2 t + π2
)needs to be
in the form f (t) = A sin [a (t + b)] + c. So write,
y (t) = 5 sin(
12t +
π
2
)= 5 sin
[12
(t + π)]
.
The fundamental cosine function is the same basic shape asthe fundamental sine function with a phase shift of π
2 units(one forth of the period) to the right. The period of the given
function is2π∣∣ 12
∣∣ = 4π and one forth of the period is π. So, sim-
ilarly shift the given function to get the given answer. Thereare other acceptable answers for this question including
ya (t) = −5 cos[12
(t + 2π)]
and yb (t) = 5 cos[12
(t − 4π)]
.
�
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
HINT
To find a sine function whose graph is iden-tical to the graph of the function y (t) =12 cos [π (t + 1)] + 1 use an appropriate phaseshift.
WebT
rigFlash
Cards
AcroTE X
eDucation
Bundle
Hint Soln
Next Home
Answer: y (t) = 12 sin
[π
(t + 3
2
)]+ 1
Solution: The fundamental sine function is the same basicshape as the fundamental cosine function with a phase shiftof π
2 units (one forth of the period) to the left. The period of
the given function is2π
π= 2 and one forth of the period is 1
2 .
So, similarly shift the given function to get the answer.
y (t) =12
sin[π
(t +
32
)]+ 1
There are other acceptable answers for this question including
ya,b (t) = ±12
sin[π
(t ∓ 1
2
)]+ 1.
�