Translating Sine and Cosine Functions

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Graphing Sine and Cosine Functions

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Key Questions

  • 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(x-4)#, we can think of it as graphing #y=f(x-4)#.
    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(x-4) [-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(x-h)# 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 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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