Trigonometry/ProOneSoftware-Trigonometry-Win31.iso/trig/chapter3.3r

Text File | 1995-04-09 | 7KB | 200 lines

 198
à 3.3ïTangent and Cotangent Graphs
äPlease give the period, phase shift, and vertical transla-
êëtion of the following tangent functions.
â
êïFind the period, phase shift, and vertical translation of
êêë yï=ï2∙tan 2(x - π/8) + 1

The period is π/2, the phase shift is π/8 units to the right, and the
vertical translation is 1 up.ï(Please see Details for an explanation.)
éSïSince the tangent is defined to be y/x, it can be expressed as
y/x = (y/r)/(x/r) = (sin x)/(cos x).ïThus, the tan x = (sin x)/(cos x).
Once again we can return to the Key Feature to learn how to graph the
basic tangent function.ïSince the sin x is in the numerator, the tan-
gent is zero whenever the sin x is zero.ïAlso, since the cos x is in
the denominator, the tangent approaches infinity whenever the cos x
approaches zero.ïThus, the tangent is zero at multiples of π, and
approaches infinity at multiples of π/2.ïAs x goes from 0 to π/2, the
tangent values steadily increase without bound.ïAs x goes from 0 to
-π/2, the tangent values steadily decrease without bound.ïThis pattern
repeats itself in the interval from π/2 to 3π/2 and so on.ïTherefore,
the period of the basic tangent function is π.ïThe basic tangent curve
is seen in the following graph.
@fig3301.bmp,350,235
The basic tangent function is periodic
with period π.ïSince it is unbounded,
there is no amplitude.ïIn general the
#period of y = A∙tan B(x + C) + D is π/│B│.
ïThe phase shift of this function is C
units to the right if C is negative and C
units to the left if C is positive.ïThe
vertical translation is D units up if D
is positive and D units down if D is negative. Please enter the phase
shift C as a real number is decimal form.
è You are encouraged to go to the Function Plotter at this time to
experiment with graphs of tangent functions using different values of A,
B, C, and D.
 1êèFind the period and phase shift of
êêêèyï=ï2∙tan 3(x - π/16)

êëA)ïperiod 3 and phase shift is π/16 to the left
êëB)ïperiod π/3 and phase shift is π/16 to the right
êëC)ïperiod 2 and phase shift is 3 units to the left
êëD)ïå of ç
ü

ëThe period is π/3 and the phase shift is π/16 units to the right.
Ç B
 2êèFind the period and phase shift of
êêêèyï=ïtan π(x - 1)

êëA)ïperiod π and phase shift is 1 to the right
êëB)ïperiod π/2 and phase shift is 1 to the left
êëC)ïperiod 1 and phase shift is 1 unit to the right
êëD)ïå of ç
ü

êThe period is 1 and the phase shift is 1 unit to the right.
Ç C
 3êèFind the period and phase shift of
êêêèyï=ï1/2∙tan (2x + π/2)

êëA)ïperiod π/2 and phase shift is π/4 to the left
êëB)ïperiod π and phase shift is π/2 to the left
êëC)ïperiod π/2 and phase shift is π/2 units to the left
êëD)ïå of ç
ü
è You should factor the 2 out of the parençs before identifying
the phase shift.ïyï=ï1/2∙tan 2(x + π/4)
êThe period is π/2 and the phase shift is π/4 units to the left.
Ç A
äPlease give the period, phase shift, and vertical transla-
êëtion of the following cotangent functions.
â
êïFind the period, phase shift, and vertical translation of
êêë yï=ï2∙cot 2(x - π/8) + 1

The period is π/2, the phase shift is π/8 units to the right, and the
vertical translation is 1 up.ï(Please see Details for an explanation.)
éSïSince the cotangent is defined to be x/y, it can be expressed as
x/y = (x/r)/(y/r) = (cos x)/(sin x).ïThus, the cot x = (cos x)/(sin x).
Once again we can return to the Key Feature to learn how to graph the
basic cotangent function.ïSince the cos x is in the numerator, the co-
tangent is zero whenever the cos x is zero.ïAlso, since the sin x is in
the denominator, the cotangent approaches infinity whenever the sin x
approaches zero.ïThus, the cotangent is zero at multiples of π/2, and
approaches infinity at multiples of π.ïAs x goes from π/2 to 0, the
tangent values steadily increase without bound.ïAs x goes from π/2 to
0, the cotangent values steadily decrease without bound.ïThis pattern
repeats itself in the interval from π to 2π and so on.ïTherefore,
the period of the basic cotangent
function is π.ïThe basic tangent
curve is seen in the following graph.
@fig3302.bmp,300,215
ïThe basic cotangent function is
periodic with period π.ïSince it is
unbounded, there is no amplitude.ïIn
general the period of
#èyï= A∙cot B(x + C) + D is π/│B│.
The phase shift of this function is C
units to the right if C is negative
and C units to the left if C is positive. The vertical translation is D
units up if D is positive and D units down it D is negative.ïPlease
enter the phase shift C as a real number is decimal form.
è You are encouraged to go to the Function Plotter at this time to
experiment with graphs of cotangent functions using different values of
A, B, C, and D.
 4êèFind the period and phase shift of
êêêèyï=ï2∙cot 3(x - π/16)

êëA)ïperiod 3 and phase shift is π/16 to the left
êëB)ïperiod π/3 and phase shift is π/16 to the right
êëC)ïperiod 2 and phase shift is 3 units to the left
êëD)ïå of ç
ü

ëThe period is π/3 and the phase shift is π/16 units to the right.
Ç B
 5êèFind the period and phase shift of
êêêèyï=ïcot π(x - 1)

êëA)ïperiod π and phase shift is 1 to the right
êëB)ïperiod π/2 and phase shift is 1 to the left
êëC)ïperiod 1 and phase shift is 1 unit to the right
êëD)ïå of ç
ü

êThe period is 1 and the phase shift is 1 unit to the right.
Ç C
 6êèFind the period and phase shift of
êêêèyï=ï1/2∙cot (2x + π/2)

êëA)ïperiod π/2 and phase shift is π/4 to the left
êëB)ïperiod π and phase shift is π/2 to the left
êëC)ïperiod π/2 and phase shift is π/2 units to the left
êëD)ïå of ç
ü
è You should factor the 2 out of the parençs before identifying
the phase shift.ïyï=ï1/2∙cot 2(x + π/4)
êThe period is π/2 and the phase shift is π/4 units to the left.
Ç A
ä Please graph the following functions by hand.

â
êêêè Graph one period of
êêê yï=ï2∙tan 2(x - π/8) + 1

êêï(Please see Details for an explanation.)
éSïThe function yï=ï2∙tan 2(x - π/8) + 1 can be graphed on the
Function Plotter in this program.ïYou are encouraged to go to the Func-
tion Plotter at this time and graph this function.
è To graph this function by hand, you should first find the period,
the phase shift, and the vertical translation.ïThe period is π/2, the
phase shift is π/8 units to the right, and the vertical translation is
1 unit up.ïNext, you should draw a coordinate axis with one hash mark
on each side of the origin and a dotted vertical line through the hash
mark.ïYou should also enter one half of the period length on the hash
mark, and draw the basic tangent function showing the affects of the
period change.

êêêèFinally, you should shift
êêêèthis dotted curve to the
êêêèright π/8 units and up
êêêèone unit.
@fig3303.bmp,25,215
@fig3304.bmp,400,215
 7

ïGraph one period ofêëA)êêë B)
ïy = 2∙tan 3(x - π/16)

@fig3305.bmp,420,5
@fig3306.bmp,525,5
üêêêïThe period is π/3 and the phase shift
êêêê is π/16 units to the right.





@fig3305.bmp,125,145
Ç A
 8

èGraph one period ofë A)êêêïB)
èy = cot π(x - 1)

@fig3307.bmp,420,5
@fig3308.bmp,525,5
üêêêêïThe period is 1 and the phase
êêêêê shift is 1 unit to the right.





@fig3308.bmp,125,145
Ç B