6.5 Graphing Sine and Cosine Functions with Phase Shifts
A phase shift on the function causes the graph to shift horizontally. By setting the
angular expression equal to zero and solve will tell you which way and how far to shift
the graph of the function. For instance, y = sin (x + π/2) has a phase shift because of the
angular expression presented in parentheses.
x + π/2 = 0
x = - π/2 represents shifting in the negative direction,
left, π/2 unit.
Example Graph one period of y = cos (x – π)
x–π=0
x = π (positive shift)
Try the following:
y = sin (x – π/4)
y = cos (x + π/2)
Answers:
The graphs of sine and cosine functions can also exhibit a vertical shift. A vertical shift
is determined by what is added, or subtracted, to the trig function.
For instance, y = sin x + 2 has a vertical shift up 2 units, while y = sin x – 1 has vertical
shift down 1 unit.
Example Determine what kind of shift occurs and which direction it goes. Graph one
cycle of the function y = sin x – 2
Vertical shift down 2 units
Try the following:
Determine what kind of shift occurs and which direction it goes. Graph one cycle of the
functions:
y = cos x + 1
y = sin x + 3
y = cos x – 4
Answers:
Vertical shift up 1 unit
Vertical shift up 3 units
Vertical shift down 4 units
To graph a trigonometric function with multiple transformations an order of operations is
required. Various methods are said to be the order of transformations. In a nut shell, the
vertical shift and horizontal shift are the last transformations to perform. It does not
matter which shit you do last.
Example
Graph one period of y = -3sin (x + π/2)
Vertical stretch factor of 3
Reflect across x-axis
Horizontal shift left π/2 unit
Example
Graph one period of y = cos (2x – π) + 3
Period: 2π/2 = π
Vertical shift up 3 units
Horizontal shift right π/2 unit
Try the following:
Graph one period.
y = -sin ( ½ x + π/4)
y = 2cos x – 4
y = cos (x + π/2) – 1
y = -4sin (2x + π) + 3
Answers:
Period: 2π/ ½ = 4π
Reflect across x-axis
Horizontal shift left π/2 unit
Period: 2π/1 = 2π
Vertical stretch factor 2
Vertical shift down 4 units
Period: 2π/1 = 2π
Vertical shift down 1 unit
Horizontal shift left π/2 unit
Period: 2π/2 = π
Reflect across x-axis
Vertical stretch factor 4
Vertical shift up 3 units
Horizontal shift left π/2 unit
My recommended method for transforming the trigonometric functions:
1. accommodate period of trig function
2. vertically stretch and/or reflect
3. shift vertically and/or horizontally