Trigonometric Graphs

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 or number to which the function is applied.

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

$\sin\left(\theta+\frac{\pi}{4}\right)$`s``i``n`(`θ`+π4)

$\cos(x-0.5)$`c``o``s`(`x`−0.5)

$\tan\left(\alpha+\frac{22}{7}\right)$`t``a``n`(`α`+227)

The following graph shows the functions $\cos x$`c``o``s``x` and $\cos(x+0.4)$`c``o``s`(`x`+0.4) on the same axes.

The graph of $\cos x$`c``o``s``x` is shown in black. It can be seen that the graph of $\cos(x+0.4)$`c``o``s`(`x`+0.4) is the graph of $\cos x$`c``o``s``x` 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 $\cos(x+0.4)$`c``o``s`(`x`+0.4) when $x=0.5$`x`=0.5 attains the same value reached by $\cos x$`c``o``s``x` when $x=0.9$`x`=0.9. Thus, the shift is to the left.

The following graph looks like the graph of $\sin x$`s``i``n``x` with a phase shift of $1.05$1.05 to the right.

The graph must belong to the function given by $\sin(x-1.05)$`s``i``n`(`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 $\sin\left(x-\frac{\pi}{3}\right)$`s``i``n`(`x`−π3).

Consider the given graph of $y=\cos\left(x+\frac{\pi}{2}\right)$`y`=`c``o``s`(`x`+π2).

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What is the amplitude of the function?

How can the graph of $y=\cos x$

`y`=`c``o``s``x`be transformed into the graph of $y=\cos\left(x+\frac{\pi}{2}\right)$`y`=`c``o``s`(`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.

EBy 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

Consider the function $f\left(x\right)=\cos x$`f`(`x`)=`c``o``s``x` and $g\left(x\right)=\cos\left(x-\frac{\pi}{2}\right)$`g`(`x`)=`c``o``s`(`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

Dvertical 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...

Determine the equation of the graphed function given that it is of the form $y=\cos\left(x-c\right)$`y`=`c``o``s`(`x`−`c`), where $c$`c` is the least positive value.

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Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions