Part 3/6:
This transformation takes every output of the original cosine function and doubles it. Thus, at ( x = 0 ), where ( \cos(0) = 1 ), the new function gives ( y_1 = 2 ). Consequently, the graph of ( 2 \cos(x) ) peaks at 2 instead of 1.
Vertical Compression
Conversely, vertical compression takes place when we apply the transformation:
[
y_2 = \frac{1}{c} \cdot \cos(x)
]
Using ( c = 2 ) again as an example, we find:
[
y_2 = \frac{1}{2} \cdot \cos(x)
]
This transformation reduces the maximum height of the graph, so the peaks now reach ( \frac{1}{2} ) instead of 1. The behavior around ( x = 0 ) and other key points on the cosine graph experiences a similar downward shift.