To determine the amplitude of the function [tex]\( y = 2 \cos x \)[/tex], you need to understand the general form of a cosine function. The general form of a cosine function is:
[tex]\[ y = a \cos(bx + c) + d \][/tex]
In this general form, the amplitude is given by the absolute value of the coefficient [tex]\( a \)[/tex].
For the function [tex]\( y = 2 \cos x \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is [tex]\( 2 \)[/tex].
Therefore, the amplitude is the absolute value of [tex]\( 2 \)[/tex]:
[tex]\[ \text{Amplitude} = |2| = 2 \][/tex]
So, the amplitude of the function [tex]\( y = 2 \cos x \)[/tex] is [tex]\( 2 \)[/tex].
For the graph:
- The amplitude indicates the distance from the midline (or the horizontal axis if there is no vertical shift) to the maximum or minimum value of the function.
- Since the amplitude is [tex]\( 2 \)[/tex], the graph will oscillate between [tex]\( 2 \)[/tex] and [tex]\( -2 \)[/tex].
- The cosine function [tex]\( \cos x \)[/tex] starts at its maximum value when [tex]\( x = 0 \)[/tex], so the graph of [tex]\( y = 2 \cos x \)[/tex] will start at [tex]\( y = 2 \)[/tex] when [tex]\( x = 0 \)[/tex] and will exhibit periodic behavior with peaks at [tex]\( y = 2 \)[/tex] and troughs at [tex]\( y = -2 \)[/tex].
In summary, the amplitude of the function [tex]\( y = 2 \cos x \)[/tex] is [tex]\( 2 \)[/tex], and this determines that the cosine wave will oscillate between [tex]\( 2 \)[/tex] and [tex]\( -2 \)[/tex].
