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How to Find the Phase Shift of a Cosine Function

Imagine a circle with a single horizontal line dividing the circle in two halves. Now imagine a small dot moving around the circumference of the circle. As the dot moves, its distance from the center of the circle, as measured along the horizontal line, increases and decreases in a periodic fashion. This is a visual image of a cosine function. Mathematicians and researchers use cosine and sine functions to model periodic processes in the world. The phase shift of a cosine function, or the function's left- or rightward displacement when you graph it, is an essential dimension of this modeling. You can easily determine the phase shift of a cosine function.

Instructions

Example 1
- 1
  Write the cosine function you want to analyze. For example, y = cos(x - 7).
- 2
  Look at the value following the x-variable within the parentheses. Determine whether the value following the x is positive or negative. For example, the value following the x-variable is negative (- 7).
- 3
  Write the value following the x-variable. If the value after the variable is negative, the phase shift of the cosine function will be to the right exactly the number of units represented by that value. For example, write "- 7." The phase shift of the cosine function is seven units to the right.
Example 2
- 4
  Write another cosine function you want to analyze. For example, y = cos(x + 1).
- 5
  Look at the value following the x-variable inside the parentheses. Determine whether the value following the x is positive or negative. For example, the value following the x-variable is positive (+ 1).
- 6
  Write the value after the x-variable. Note that if the value after the x-variable is positive, the phase shift of the cosine function will be to the left exactly the number of units represented by that value. For example, write "+ 1." The phase shift of the cosine function is one unit to the left.

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