To complete the given statements:
1. The function [tex]\( y = \cos(2x) - 2 \)[/tex] has the same amplitude as the function [tex]\( y = \cos(x - 4) + 1 \)[/tex].
Step-by-step explanation:
- For any cosine function of the form [tex]\( y = a \cos(bx - c) + d \)[/tex], the amplitude is given by the absolute value of the coefficient [tex]\( a \)[/tex]. So for both given functions:
- [tex]\( y = \cos(2x) - 2 \)[/tex]: Here [tex]\( a = 1 \)[/tex], so the amplitude is [tex]\( 1 \)[/tex].
- [tex]\( y = \cos(x - 4) + 1 \)[/tex]: Here [tex]\( a = 1 \)[/tex], so the amplitude is [tex]\( 1 \)[/tex].
- Therefore, both functions have the same amplitude, which is [tex]\( 1 \)[/tex].
2. The function [tex]\( y = \cos(-2x) \)[/tex] is a translation of the parent cosine function.
Step-by-step explanation:
- Considering the function [tex]\( y = \cos(-2x) \)[/tex]:
- [tex]\( y = \cos(-2x) \)[/tex] is essentially the same as [tex]\( y = \cos(2x) \)[/tex] because the cosine function is an even function, meaning [tex]\( \cos(-\theta) = \cos(\theta) \)[/tex].
- Therefore, [tex]\( y = \cos(2x) \)[/tex] and [tex]\( y = \cos(x) \)[/tex] can be compared.
- The transformation [tex]\( y = \cos(2x) \)[/tex] is a horizontal compression by a factor of [tex]\( 2 \)[/tex] compared to the parent function [tex]\( y = \cos(x) \)[/tex], but no vertical or horizontal shift is present beyond this compression.
Thus, combining the explanations:
- The amplitudes of [tex]\( y = \cos(2x) - 2 \)[/tex] and [tex]\( y = \cos(x - 4) + 1 \)[/tex] are both [tex]\( 1 \)[/tex], so they have the same amplitude.
- The function [tex]\( y = \cos(-2x) \)[/tex] does not involve a translation but rather a compression relative to the parent function, thus it translates as "same as the parent function" with respect to no horizontal or vertical shifts beyond compression.
Therefore, the completed statements are:
1. The function [tex]\( y = \cos(2x) - 2 \)[/tex] has the same amplitude as the function [tex]\( y = \cos(x - 4) + 1 \)[/tex].
2. The function [tex]\( y = \cos(-2x) \)[/tex] is the same as the parent function with respect to no horizontal or vertical shifts beyond compression.
