Translating Sine and Cosine Functions

Key Questions

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

    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!

    Antoine · 1 · Apr 12 2015
  • How do you graph sine and cosine functions when it is translated?

    I think you'll find a useful answer here: http://socratic.org/trigonometry/graphing-trigonometric-functions/translating-sine-and-cosine-functions

    Vertical translation

    Graphing y=sinx+k Which is the same as y=k+sinx:

    In this case we start with a number (or angle) x. We find the sine of x, which will be a number between -1 and 1. The after that, we get y by adding k. (Remember that k could be negative.)

    This gives us a final y value betwee -1+k and 1+k.

    This will translate the graph up if k is positive (k>0)
    or down if k is negative (k<0)

    Examples:

    y=sinx
    graph{y=sinx [-5.578, 5.52, -1.46, 4.09]}

    y=sinx+2 = 2+sinx
    graph{y=sinx+2 [-5.578, 5.52, -1.46, 4.09]}

    y=sinx-4=-4+sinx
    graph{y=sinx-4 [-5.58, 5.52, -5.17, 0.38]}

    The reasoning is the same for y=cosx+k=k+cosx, but the starting graph looks different, so the final graph is also different:

    y=cosx
    graph{y=cosx [-5.578, 5.52, -1.46, 4.09]}

    y=cosx+2 = 2+cosx
    graph{y=cosx+2 [-5.578, 5.52, -1.46, 4.09]}

    Jim H · 1 · Mar 22 2015

Questions

Graphing Trigonometric Functions