Lesson

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 to which the function is applied.

For example, the following functions include a phase shift transformation.

$\sin\left(\theta+45^\circ\right)$`s``i``n`(`θ`+45°)

$\cos(x-37.5^\circ)$`c``o``s`(`x`−37.5°)

$\tan\left(\alpha+180^\circ\right)$`t``a``n`(`α`+180°)

The following graph shows the functions $\cos\theta$`c``o``s``θ` and $\cos(\theta+23^\circ)$`c``o``s`(`θ`+23°) on the same axes.

The graph of $\cos\theta$`c``o``s``θ` is shown in black. It can be seen that the graph of $\cos(\theta+23^\circ)$`c``o``s`(`θ`+23°) is the graph of $\cos\theta$`c``o``s``θ` shifted to the *left *by the amount $23^\circ$23°.

The dotted lines drawn on the diagram are intended to show that the function $\cos(\theta+23^\circ)$`c``o``s`(`θ`+23°) when $\theta=20^\circ$`θ`=20° attains the same value reached by $\cos\theta$`c``o``s``θ` when $\theta=43^\circ$`θ`=43°. Thus, the shift is to the left.

The following graph looks like the graph of $\sin\theta$`s``i``n``θ` with a phase shift of $60^\circ$60° to the right.

The graph must belong to the function given by $\sin(\theta-60^\circ)$`s``i``n`(`θ`−60°). The phase shift to the right has been brought about by adding $-60^\circ$−60° to $\theta$`θ`.

Consider the function $f\left(x\right)=\sin x$`f`(`x`)=`s``i``n``x` and $g\left(x\right)=\sin\left(x-90^\circ\right)$`g`(`x`)=`s``i``n`(`x`−90°).

Complete the table of values for both functions.

$x$ `x`$0$0 $90^\circ$90° $180^\circ$180° $270^\circ$270° $360^\circ$360° $f\left(x\right)$ `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`)?Horizontal translation $90^\circ$90° to the right

AHorizontal translation $90^\circ$90° to the left

BVertical translation $90^\circ$90° upwards

CVertical translation $90^\circ$90° downwards

DHorizontal translation $90^\circ$90° to the right

AHorizontal translation $90^\circ$90° to the left

BVertical translation $90^\circ$90° upwards

CVertical translation $90^\circ$90° downwards

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...

The functions $f\left(x\right)$`f`(`x`) and $g\left(x\right)=f\left(x+k\right)$`g`(`x`)=`f`(`x`+`k`) have been graphed on the same set of axes in grey and black respectively.

Loading Graph...

What transformation has occurred from $f\left(x\right)$

`f`(`x`) to $g\left(x\right)$`g`(`x`)?Horizontal translation of $135^\circ$135° left.

AHorizontal stretching by a factor of $135^\circ$135°.

BHorizontal translation of $135^\circ$135° right.

CVertical translation of $135^\circ$135° up.

DHorizontal translation of $135^\circ$135° left.

AHorizontal stretching by a factor of $135^\circ$135°.

BHorizontal translation of $135^\circ$135° right.

CVertical translation of $135^\circ$135° up.

DDetermine the smallest positive value of $k$

`k`.

The graph of $y=\sin x$`y`=`s``i``n``x` is translated $60^\circ$60° to the left.

What is the equation of the new curve?

What is the amplitude of the new curve?

What is the period of the new curve?