Translating Sine and Cosine Functions
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Horizontal Translation
One way to think about horizontal translations of a function is to think about the value of
#x# that will cause us to find#f(0)# .We know the graph of
#y=f(x)=sin(x)# .To graph
#y=sin(x4)# , we can think of it as graphing#y=f(x4)# .
Now, what value of#x# will make me find#f(0)=sin(0)# ? Clearly, it is#x=4# .
So "#4# is the new#0# ". Everything moves#4# to the right.graph{y=sin(x4) [0.498, 7.295, 2.302, 1.596]}
To graph
#y=sin(x+ pi/3)# , we ask ourselves, "What value of#x# will cause us to find#sin(0)# ?
That will be#x= pi/3#
So, "# pi/3# is the new#0# ". Everything moves# pi/3# to the right. Wait a minute, surly it's more clear to say: Everything moves#pi/3# to the left.(For the graph below, remember that
#pi/3# is a little more than#1# )graph{y=sin(x+pi/3) [3.02, 1.845, 1.192, 1.241]}
To start the graph of
#y=sin(bx c)# , we'll need to change the period and also translate.Cosine
The reasoning for graphing
#y=cos(xh)# is the same. The difference is that
#sin(0)=0# , so the point corresponding to the 'new#0# ' goes on the#x# axis
#cos0=1# , so the point corresponding to the 'new#0# ' goes at#y=1# #y = cos (x+ pi/3)#
graph{y=cos(x+pi/3) [3.02, 1.845, 1.192, 1.241]} 
For an equation:
A vertical translation is of the form:
#y = sin(theta) + A# where# A!=0#
OR#y = cos(theta) + A# Example:
#y = sin(theta) + 5# is a#sin# graph that has been shifted up by 5 unitsThe graph
#y = cos(theta)  1# is a graph of#cos# shifted down the yaxis by 1 unitA horizontal translation is of the form:
#y = sin(theta + A)# where#A!=0# Examples:
The graph#y = sin(theta + pi/2)# is a graph of#sin# that has been shifted#pi/2# radians to the rightFor a graph:
I'm to illustrate with an example given above:For compare:
#y = cos(theta)#
graph{cosx [5.325, 6.675, 5.16, 4.84]}and
#y = cos(theta)  1#
graph{cosx 1 [5.325, 6.675, 5.16, 4.84]}
To verify that the graph of#y = cos(theta)  1# is a vertical translation, if you look on the graph,the point where
#theta = 0# is no more at#y = 1# it is now at# y = 0# That is, the original graph of
#y= costheta# has been shifted down by 1 unit.Another way to look at it is to see that, every point has been brought down 1 unit!

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