Phase shift for trigonometric functions means moving the graph of the function to the right or to the left. This transformation occurs when a constant is added to (or subtracted from) the angle or number to which the function is applied.
For example, the following functions include a phase shift transformation.
$\sin\left(\theta+\frac{\pi}{4}\right)$sin(θ+π4)
$\cos(x-0.5)$cos(x−0.5)
$\tan\left(\alpha+\frac{22}{7}\right)$tan(α+227)
Example 1
The following graph shows the functions $\cos x$cosx and $\cos(x+0.4)$cos(x+0.4) on the same axes.
The graph of $\cos x$cosx is shown in black. It can be seen that the graph of $\cos(x+0.4)$cos(x+0.4) is the graph of $\cos x$cosx shifted to the left by the amount $0.4$0.4.
The dotted lines drawn on the diagram are intended to show that the function $\cos(x+0.4)$cos(x+0.4) when $x=0.5$x=0.5 attains the same value reached by $\cos x$cosx when $x=0.9$x=0.9. Thus, the shift is to the left.
Example 2
The following graph looks like the graph of $\sin x$sinx with a phase shift of $1.05$1.05 to the right.
The graph must belong to the function given by $\sin(x-1.05)$sin(x−1.05). The phase shift to the right has been brought about by adding $-1.05$−1.05 to $x$x.
A more precisely drawn horizontal scale might reveal that the graph actually crosses the axis at $1.047$1.047 which is approximately $\frac{\pi}{3}$π3. So, another way of writing the function is with the expression $\sin\left(x-\frac{\pi}{3}\right)$sin(x−π3).
Worked examples
Question 1
Consider the given graph of $y=\cos\left(x+\frac{\pi}{2}\right)$y=cos(x+π2).
What is the amplitude of the function?
How can the graph of $y=\cos x$y=cosx be transformed into the graph of $y=\cos\left(x+\frac{\pi}{2}\right)$y=cos(x+π2)?
By reflecting it about the $x$x-axis, and then translating it horizontally $\frac{\pi}{2}$π2 units to the left.
ABy reflecting it about the $x$x-axis, and then translating it horizontally $\frac{\pi}{2}$π2 units to the right.
BBy translating it horizontally $\frac{\pi}{2}$π2 units to the right.
CBy changing the period of the function.
DBy translating it horizontally $\frac{\pi}{2}$π2 units to the left.
E
Question 2
Consider the function $f\left(x\right)=\cos x$f(x)=cosx and $g\left(x\right)=\cos\left(x-\frac{\pi}{2}\right)$g(x)=cos(x−π2).
Complete the table of values for both functions.
$x$x $0$0 $\frac{\pi}{2}$π2 $\pi$π $\frac{3\pi}{2}$3π2 $2\pi$2π $f\left(x\right)$f(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $g\left(x\right)$g(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Using the table of values, what transformation of the graph of $f\left(x\right)$f(x) results in the graph of $g\left(x\right)$g(x)?
vertical translation $\frac{\pi}{2}$π2 units downwards
Ahorizontal translation $\frac{\pi}{2}$π2 units to the left
Bhorizontal translation $\frac{\pi}{2}$π2 units to the right
Cvertical translation $\frac{\pi}{2}$π2 units upwards
DThe graph of $f\left(x\right)$f(x) has been provided below.
By moving the points, graph $g\left(x\right)$g(x).
Loading Graph...
Question 3
Determine the equation of the graphed function given that it is of the form $y=\cos\left(x-c\right)$y=cos(x−c), where $c$c is the least positive value.