Answered

From everyday questions to specialized queries, IDNLearn.com has the answers. Whether it's a simple query or a complex problem, our community has the answers you need.

Use the drop-down menus to complete each statement.

The function [tex]y = \cos(2x) - 2 \quad v[/tex] has the same amplitude as the function [tex]y = \cos(x - 4) + 1[/tex].

The function [tex]y = \cos(-2x) \quad v[/tex] is a translation of the parent cosine function.

Sagot :

GinnyAnsweridn
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.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.