College Algebra & Trigonometry, 2018a

9.4. PHASE SHIFT 417
9.4 Phase Shift
The last form of transformation we will discuss in the graphing of trigonometric
functions is the phase shift, or horizontal displacement. So far, we have considered
the amplitude, period and vertical shift transformations of trigonometric
functions. In the standard equation y = A sin(Bx)+D, these corrrespond to the
coefficients A, B and D. Notice that the amplitude and vertical shift coefficients
(A and D), which affect the y-axis occur outside of the trigonometric function,
whereas the coefficient that affects the period of the graph along the x-axis occurs
within the sine function. This is true of the phase shift as well.
If we consider a general equation of:
y = A sin(Bx + C)+D
the constant C will affect the phase shift, or horizontal displacement of the function.
Let’s look at a simple example.
Example 1
Graph at least one period of the given function: y = sin(x + π).
Be sure to indicate important points along the x and y axes.
Let’s examine this function by looking at a table of values.
x x + π sin(x + π)
0 π 0
π/2 3π/2 −1
π 2π 0
3π/2 5π/2 1
2π 3π 0