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 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.
|
#y=# #\cos \left(x\right)-4#
In step 1 we saw that the green graph is obtained by shifting the blue graph downwards by #4#. Hence, we subtract #4# from the formula of the blue graph #y=\cos \left(x\right)#. This gives us the following formula for the green graph:
\[y=\cos \left(x\right)-4\]
Or visit omptest.org if jou are taking an OMPT exam.