EOUATIONS OF SINUSOIDAL FI]NCIIONS
Inve*tigatlng the Equatiore of Sinusoidd Functioue:
y--o,,sin(by+c)+
A. Graphy
=rinr
for the dom4in
cI
{xl-2r Sx S3tr, x e R}.
Then gaph the following funetions on the sarDe arGs.
i' Y:3 cinx
ii. Y: 5 sinx
iii. Y: % stnt
Ss,l x
B, Compare your graphs in part A. How
y:sinx?
does changing the value
c^V:g.nge5 *h.e
ofa affect the graph of
arnpl rl-ude
u
C. Graph, in radian mode, y = sln *, for the domaia {xl
-2r <x
< 3q x e R}.
Then graph the following fl.rnctiors on &e same ores.
i.
Y
= sid2x
'lj.y=sin3x
iii.
Y
-
t;n Y",
D. Comparc yorn graphs in port C. How
y:
changitg the value of6 affect the graph
of
sini?
C
harrrys
(
?<t'tod
lnet< i^.-og
l-x-rr-'r-z <:rr-l
f"^ f fg )
E. Graph, in radian mode,Y = sinx,for
Then graph the following functions on
domain {xl
-2t
<x<
3r, x € R).
salne axes,
i. y=sinx+3
ii.
y:
sin
x- I
iii. y:sinx+4
\/-Sl..x+t/
{,n",s
tf
F. Compare your graPhs in Part G. How does changing the value
y = sin x2
Lhomq<s
*te rnrd \rn<-
ofd affect the graph of
r
G. Graph, in radian mode,y: srn.r, for the domain {xl-22 < x < 3r, x e R}.
Then graph the following functions on the same axes.
i. y = sin (x-
tr)
ii. y=sin(x+ t)
Srrrr (1..r11
H. Compare your graphs in part E. How does changing the value of c affect the graph
y=sinx?
-,,,-]-:tr, r,
2
;';-|cul
Sl-r r {-1
of
;f
Reflectine:
I.
Consider the sinusoidal equation:
)-
a sr'n lbx+c)+
Explain how to determine each ofthe following characteristics ofthe function, based
on the values ofa, D, and d.
o
tlre
r
the equation ofthe
o
the period
amplitude -_ A_-
midline = d
l-ll
b
J.
=
o
maximum
point
d n a-
o
minimum
point
Cl- n
Consider the
equation:
PZJET)
y:asin(x-c)+d
How does the value ofc affect the graph
tra.s -cr'p)--as€ sh'{}
iF c+ Ot Jqrc-p'6
--l
[r,o, zo,-iot)
27
[r.
I
Consider the function
1':2cos4x+l
for {xl 05x<360'.xeR}.
Describe the graph ofthe function by stating the amplitude, equation ofthe midline, range,
and period.
l-r-'\o.X- \r2--<
Ct.}.
.-l
LI
Y\\\'\
l
Your
tu
{xi
'- 3,9c 1e
\)
<, , ct\ -- ,1H-q
a-f-)
.:
ln:
Consider the function
for
I
I
K )l- I '
rl
I 2
y:5cos%x-3
0<x<360',xeR}.
Describe the graph ofthe function by stating the amplitude, equation ofthe midline, range,
and period.
c\> b
d--3
A". ,)9l-S-9r2,1trz
Pe
'
ror-\= 1]I
t)
-.lTf
2
28
Ex. 2
y:3sin2(x-45)
Consider the function
a.
Describe the graph ofthe function by stating the amplitude, equation ofthe midline,
range, and period, as well as tIe relevant horizontal translation ofy = srn x.
ot=
3
F\.-i
d=o
LlS" r,
R-yl - ?>9 1j ,X((
11gr
b. How would
.hF-
'od = Ztr =-W
the graph of
gh
y:3cos 2(x+ 4.!? bethe same? How would it be
different?
a-= 3
HT LlSo le{-+
d=a
\
)l
?<.a
--
tod ;
.-.:
Z-n-
saJr Z
Your turn:
y:4cos3(x-607
Consider the function:
Describe the graph ofthe function by stating the amplitude, equation ofthe midline, range,
cos .r.
and period, as well as the relevant horizontal translation ofy
:
0t
"l
d,O
w
iu,c).-_
1I
a
Go" c 5[t1
'
R.'. )l-q'- g'- .1
29
f,r.
3
The Far North is called ..the
Land,of the Midnight Sun,, for good
a
reason: during the
summer months, in some locarions,
th.
i*ir"rr"
i o"r. The number
hours of daylight in Iqaluit,
"u,iu" "i.iul"
s*
Nunavut,._
y
where
r
:
b".6;.;;;;;;;Hi."
8.245 sin 0.0172(x
_
80.988)
+
of
12.585
is the day number in the year.
a. How many hours ofdaylight occur
;n Iqaluit
i.
the shortest day ofthe year?
on the following days?
rnrn = cl_a_
= 11.58S_
ii.
the rongestday of the
r-11'rr-,
year? = Q 39 hoc'rs -* l t,/'"n n
v\-b-x, c-Lr-c1
'
- \ )-=6 S i B,z..ts
b. In some years, June 2 r is the rongest
d1r:.***
I I.0l pm on June 2 t. At what
riire did,h.
i"
t",
-11,"i;;";;*
^
;f
di;;;;.- "-,, "",.
ll'.ol
t\:
g.A\a
1ro,
p,-n -> Se,rrfrs(, (- aDh, {i.Lt
gtl afY1
-) \Z huur> er.!-rlr_r r
rnrn
a(r.rtrri(-
?.C:\clrn
2el nCt,rs ecWlrJ
-)
1.llan1
.
_.
)
a r".\
So
rnrh
?Cw tr((
?,9 1".I, ,
c. What is the period of rhis sinur;ordal
lilnction? Explain how the period
relates to the
context orthe problem.
p.-,oc.i_- Z_y
o.orr1 -- 5G5'3 da,.,s /1n^,'
d. What does the value ofc,
Ot(J
g0.9gg, represent
in the context ofthe problem?
c(rt o{ tht y<rr (ffir-c,t"}}-.)
30
Your Turn:
graphed witi a thin line'
ln each diagram, the sinusoidal function ) = sin t has been
differs from rhe graph
dtick-lin;
*j$
ihc
gropn"a
tun"rio,
of r,= sin r bv
. a change in pcriod
' a change in amplitude
. a vertical shift in the midline.
. a horizontal phase shift
B;.'##iiiil;;;;;;iaui
There rnay be more than one changc involved
b)
.\'
5
(\.-
7-,"d
hc,.
p s
\-\
p*.r,
tbd
h ps
vt
-o-
W{'oo
'vl
3l