How do you identify the vertical and horizontal translations of sine and cosine from a graph and an equation?

1 Answer
Antoine
Apr 12, 2015

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 units

The graph #y = cos(theta) - 1# is a graph of #cos# shifted down the y-axis by 1 unit

A 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 right

For 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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