Trigonometry/ProOneSoftware-Trigonometry-Win31.iso/trig/chapter3.2r

Text File | 1995-04-09 | 10KB | 366 lines

 364
à 3.2ïGraphing Cosine Functions
äPlease find the amplitude of the following cosine functions.
â
êêêèFind the amplitude of
êêê yï=ï3∙cos 4(x - π/4) + 1.

êêïThe amplitude of this cosine function is 3.
êêè(Please see Details for an explanation.)
éSïYou are encouraged at this point to study the Key Feature in
this program very carefully to understand how the curve, y = cos Θ, is
generated.
êêêè Let Θ be any real number.ïThen, Θ in the fig-
êêê ure is both the radian measure of the central an-
êêê gle and the real number length of the arc subten-
êêê ed on the unit circle by the central angle.ïThe
êêê value of the function, y = cos Θ, at the real num-
êêê ber Θ (arc length), is defined to be the same val-
@fig3101.bmp,25,88
êêê ue as the cosine of the "central angle Θ" which is
êêê x/r or in this case just x since r is 1.
è The Key Feature in this program shows one complete revolution or pe-
riod of the function, y = cos Θ.ïNegative values of Θ and values with
more than one revolution allow Θ to range through all of the real num-
bers generating the following graph.


êêêêè (more...)



@fig3202.bmp,125,235
ïYou can see in this figure that the highest value assumed by the basic
cosine curve is 1.ïThis is the amplitude of the curve.ïThe curve, y =
2∙cos Θ, has amplitude equal to 2, and y = 3∙cos Θ has amplitude equal
to 3.ïIn general the amplitude of the curve, y = A∙cos B(Θ + C) + D, is
always going to be the absolute value of A.ïThus, the amplitude can be
found by inspection.
è You are now encouraged to go to the Function Plotter in this program
and experiment graphing y = A∙cos Θ using different values of A to ob-
serve the changes in amplitude caused by changing the value of A.
êêêïBasic Cosine curve y = cos Θ









 1
êêêè Find the amplitude of

êêêïyï=ï3∙cos 2(Θ - π/8) + 1

ê A)ï3êèB)ïπêïC)ïπ/8 rightêD)ï1 up
ü

êêêè The amplitude is 3.
Ç A
 2
êêêè Find the amplitude of

êêêïyï=ï2∙cos 3(Θ - π/3) - 2

ê A)ï2π/3êB)ïπ/3 rightëC)ï2êëD)ï2 down
ü

êêêè The amplitude is 2.
Ç C
 3
êêêè Find the amplitude of

êêêïyï=ï-2∙cos (x + π/2) + 1

ê A)ïπ/2 leftë B)ï1 upêC)ï2πêëD)ï2
ü

êêêè The amplitude is 2.
Ç D
 4
êêêè Find the amplitude of

êêêïyï=ï1/2∙cos (πx - π/4) - 1

ê A)ï1 downëB)ï1/2êïC)ï2êèD)ï1/4 right
ü

êêêè The amplitude is 1/2.
Ç B
äïPlease find the period of the following cosine functions.
â
êêêëFind the period of
êêê yï=ï3∙cos 4(x - π/4) + 1.

êë The period of this cosine function is 2π/4ï=ïπ/2.
êêè(Please see Details for an explanation.)
éSïIn the first Details of this section, the basic cosine curve was
generated.êêêïy = cos Θ






@fig3202.bmp,125,45

èThe basic cosine curve is seen to be periodic, that is it does the
same thing over and over.ïIf you choose any value of Θ and go 2π units
to the right (or left), then cos Θï=ïcos (Θ + 2π).ïThis means that
the basic cosine curve is periodic with period 2π.
è If you look at the graph of y = cos 2∙Θ, you see that the fundamen-
tal period length is 2π/2 = π.ïAlso, if you look at the graph of y =
cos 3∙Θ, you see that the fundamental period length is 2π/3.ïIn general
the period of the curve, y = A∙cos B(Θ + C) + D, is always going to be
#2π/│B│.ïThus, the period can be found by dividing B into 2π.
è You are now encouraged to go to the Function Plotter in this program
and experiment graphing y = cos B∙Θ using different values of B to ob-
serve the changes in period caused by changing the value of B.
 5
êêêë Find the period of

êêêïyï=ï3∙cos 2(Θ - π/8) + 1

ê A)ï3êèB)ïπêïC)ïπ/8 rightêD)ï1 up
ü

êêêè The period is 2π/2 = π.
Ç B
 6
êêêëFind the period of

êêêïyï=ï2∙cos 3(Θ - π/3) - 2

ê A)ï2π/3êB)ïπ/3 rightëC)ï2êëD)ï2 down
ü

êêêè The period is 2π/3.
Ç A
 7
êêêë Find the period of

êêêïyï=ï-2∙cos (x + π/2) + 1

ê A)ïπ/2 leftë B)ï1 upêC)ï2πêëD)ï2
ü

êêêè The period is 2π/1 = 2π.
Ç C
 8
êêêë Find the period of

êêêïyï=ï1/2∙cos (πx - π/4) - 1

ê A)ï1 downëB)ï1/2êïC)ï2êèD)ï1/4 right
ü

êêêè The period is 2π/π = 2.
Ç C
äïPlease find the phase shift of the following cosine func-
êêtions.
â
êêêèFind the phase shift of
êêê yï=ï3∙cos 4(x - π/4) + 1.

ëThe phase shift of this cosine function is π/4 units to the right.
êêè(Please see Details for an explanation.)
éSïIn the first Details of this section, the basic cosine curve was
generated.êêêèy = cos Θ






@fig3202.bmp,125,45

èThe basic cosine curve has zero phase shift.ïThat means that it has
not been shifted to the right or to the left.ïIf you look at the graph
of y = cos (Θ - π/4), you see that it looks like the basic cosine curve
except that it has been shifted to the right π/4 units.ïThe curve
y = cos (Θ + π/3) is seen to be shifted π/3 units to the left.ïIn gen-
eral the curve y = A∙cos B(Θ + C) + D, is shifted to the right C units
if C is negative and C units to the left if C is positive.
è You are now encouraged to go to the Function Plotter in this program
and experiment graphing y = cos (Θ + C) using different values of C to
observe the changes in phase shift caused by changing the value of C.
Please enter the values of C as real numbers in decimal form.ïFor ex-
ample, π/4 would be entered as .7854.
 9
êêêè Find the phase shift of

êêêïyï=ï3∙cos 2(Θ - π/8) + 1

ê A)ï3êèB)ïπêïC)ïπ/8 rightêD)ï1 up
ü

êêêThe phase shift is π/8 to the right.
Ç C
 10
êêêèFind the phase shift of

êêêïyï=ï2∙cos 3(Θ - π/3) - 2

ê A)ï2π/3êB)ïπ/3 rightëC)ï2êëD)ï2 down
ü

êêêThe phase shift is π/3 to the right.
Ç B
 11
êêêè Find the phase shift of

êêêïyï=ï-2∙cos (x + π/2) + 1

ê A)ïπ/2 leftë B)ï1 upêC)ï2πêëD)ï2
ü

êêëThe phase shift is π/2 units to the left.
Ç A
 12
êêêè Find the phase shift of

êêêïyï=ï1/2∙cos (πx - π/4) - 1

ê A)ï1 downëB)ï1/2êïC)ï2êèD)ï1/4 right
ü
ê You should factor π out of the parençs before identifying
ê the phase shift.
êêêyï=ï1/2∙cos π∙(Θ - 1/4) - 1
êêè The phase shift is 1/4 to the right.
Ç D
ä Please find the vertical translation of the following cosine
êëfunctions.
â
êêë Find the vertical translation of
êêê yï=ï3∙cos 4(x - π/4) + 1.

ë The vertical translation of this cosine function is 1 unit up.
êêè(Please see Details for an explanation.)
éSïIn the first Details of this section, the basic cosine curve was
generated.êêêèy = cos Θ






@fig3202.bmp,125,45


èThe basic cosine curve has no vertical translation since it is cen-
tered on the x-axis.ïIf you look at the graph of y = cos Θ + 1, you see
that the basic cosine curve has been translated up one unit.ïThe func-
tion y = sin Θ - 2 has been translated 2 units down.ïIn general the
translation of the curve y = A∙cos B(Θ + C) + D is D units up if D is
positive and D units down if D is negative.
è You are now encouraged to go to the Function Plotter in this program
and experiment graphing y = cos Θ + D using different values of D to
observe the changes in translation caused by changing the value of D.
 13
êêë Find the vertical translatiion of

êêêïyï=ï3∙cos 2(Θ - π/8) + 1

ê A)ï3êèB)ïπêïC)ïπ/8 rightêD)ï1 up
ü

êêêThe vertical translation is 1 up.
Ç D
 14
êêêFind the vertical translation of

êêêïyï=ï2∙cos 3(Θ - π/3) - 2

ê A)ï2π/3êB)ïπ/3 rightëC)ï2êëD)ï2 down
ü

êêêThe vertical translation is 2 down.
Ç D
 15
êêë Find the vertical translation of

êêêïyï=ï-2∙cos (x + π/2) + 1

ê A)ïπ/2 leftë B)ï1 upêC)ï2πêëD)ï2
ü

êêëThe vertical translation is 1 up.
Ç B
 16
êêêFind the vertical translation of

êêêïyï=ï1/2∙cos (πx - π/4) - 1

ê A)ï1 downëB)ï1/2êïC)ï2êèD)ï1/4 right
ü

êêè The vertical translation is 1 down.
Ç A
äïPlease graph the following cosine functions.
â
êë Graph one period of y = 3∙cos 4(Θ - π/4) + 1.

êêêè (Please see Details)
éSïThe function y = 3∙cos 4(Θ - π/4) + 1 can be graphed on the
Function Plotter in this program.ïYou are encouraged to go to the Func-
tion Plotter and graph this function at this time.
è It is possible to graph this function by using your built-in calcu-
lator, but it is very time consuming.ïJust enter the real number Θ,
subtract π/4, press the cos button, multiply by 3, and add 1 to get the
 coordinates of a point.ïYou can then plot points to get a graph.ïNot
very fast, but it does work.
è There is a fairly fast way to graph this function by hand however.
To use this fast method, you should first find the amplitude, period,
phase shift, and vertical translation of the curve.ïThese can be found
by inspection.ïThe amplitude is 3, the period is π/2, the phase shift
is π/4 to the right, and the vertical translation is 1 up.ïNext, you
should draw a coordinate axis with four hash marks on the x-axis and 2
marks on the y-axis.
@fig3103.bmp,125,300





êêêêêêêêë (more...)
Then, on the fourth hash mark, enter the value of the period length π/2.
Repeatedly multiply this number by one-half to find the values of the
other hash marks.ïAlso, you should enter the amplitude, 3, on the y-
axis.ïThen draw a dotted cosine curve showing the affects of the ampli-
tude and the period change.
@fig3204.bmp,220,95





@fig3205.bmp,125,305



Finally, you should shift this curve to the right π/4 units and up 1
unit.ïThis is the graph of one period of this function.




 17

Graph by hand one periodêïA)êêë B)
of y = 3∙cos 2(Θ - π/8) + 1.

@fig3206.bmp,425,15
@fig3207.bmp,725,15
üêêêëThe amplitude is 3, the period is
êêêêè π, the phase shift is π/8,
êêêêè and the vertical translation is 1 up.





@fig3207.bmp,125,145
Ç B