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.
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 xas

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.

#y=# #\cos \left(x\right)+3#
In step 1 we saw that the green graph is obtained by shifting the blue graph upwards by #3#. Hence, we add #3# to the formula of the blue graph #y=\cos \left(x\right)#. This gives us the following formula for the green graph:
\[y=\cos \left(x\right)+3\]
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