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Differentiate the following functions. Determine if each function is differentiable for all real values of .

  1. Compare the graph of with its parent .

    looks just like . Their periods are the same. Their amplitudes are the same. Their maximum and minimum -values are the same. Their SLOPES are the same. The only difference is their horizontal locations.

    The slopes will shift with the graph:

  2. The derivative will be a piecewise function as well. Differentiate each piece separately.

    Consider the boundary point of the derivative. Examine the two pieces at . Is the derivative continuous?