Transformations of trigonometric functions

Trigonometry: Trigonometric functions

Transformations of trigonometric functions

We have seen what the standard graphs of sine and cosine look like. We can also transform these functions.

Transformations

We can transform the functions #f(x)=\sin(x)# and #g(x)=\cos(x)# in four ways. We will show these using the sine function, but the cosine works in the same way.

	Transformations	Examples
1	We shift the graph of #f(x)=\sin(x)# up by #\green q#. The new function becomes \[f(x)=\sin(x)+\green q\] The period and amplitude of the function remain the same, but the equilibrium becomes equal to #\green q#.	Plaatje verticale translatie
2	We shift the graph of #f(x)=\sin(x)# to the right by #\blue p#. The new function becomes \[f(x)=\sin\left(x-\blue p\right)\] The period, amplitude and equilibrium remain the same. We call #\blue p# the phase shift.	Plaatje horizontale translatie
3	We multiply the graph of #f(x)=\sin(x)# by #\purple a# relative to the #x#-axis. The new function becomes \[f(x)=\purple a \sin(x)\] The period and equilibrium remain the same, but the amplitude becomes equal to #\purple{\left\| a \right\|}#. When #\purple a \lt 0#, the graph reverses. This means it first falls instead of rises. If #\purple a =- 1#, the new function is a reflection of the old function across the #x#-axis.	Plaatje vermenigvuldiging x-as
4	We multiply the graph of #f(x)=\sin(x)# by #\orange b# relative to the #y#-axis. This means we replace #x# with #\frac{1}{\orange b}x#. The new function becomes \[f(x)=\sin\left(\frac{1}{\orange b}x\right)\] The equilibrium and amplitude remain the same, but the period becomes equal to #\orange b \cdot 2 \pi#.	Plaatje vermenigvuldiging #y#-as.

More transformations

We can also perform multiple transformations in succession.

For example: #f(x)=\purple3 \sin\left(x-\blue4\right)# is at first a shift to the right by #\blue4#, and subsequently a multiplication by #\purple3# relative to the #x#-axis of the trigonometric function #f(x)=\sin(x)#.

Compose sine and cosine function

The standard sine function has the form #f(x)=\purple a \sin(\tfrac{1}{\orange b}(x-\blue p))+\green q#.

This function has period #\orange b \cdot 2 \pi#, amplitude #\purple a#, equilibrium #\green q# and phase shift #\blue p#. The latter means that at #x=\blue p# the graph of the sine function rises through the equilibrium.

The same applies to the cosine function. The standard cosine function is #f(x)=\purple a \cos(\tfrac{1}{\orange b}(x-\blue p))+\green q#.

This function has period #\orange b \cdot 2 \pi#, amplitude #\purple a#, equilibrium #\green q# and phase shift #\blue p#. The latter means that at #x=\blue p# the cosine function has a top (with #y#-value #\purple a + \green q#).

We note that we can obtain the cosine by translating the sine. Earlier, we saw that \[\cos(x)=\sin(x+\frac{\pi}{2})\]

Take another look at the two graphs.

The blue (solid) graph has formula:
\[y=\cos \left(x\right)\]
What is the formula of the green (dashed) graph?

#y=# #\cos \left(x\right)+5#

In step 1 we saw that the green graph is obtained by shifting the blue graph upwards by #5#. Hence, we add #5# to the formula of the blue graph #y=\cos \left(x\right)#. This gives us the following formula for the green graph:
\[y=\cos \left(x\right)+5\]

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