To transform the parent cosine function [tex]\( y = \cos(x) \)[/tex] into [tex]\( y = 0.35 \cos \left(8\left(x - \frac{\pi}{4}\right)\right) \)[/tex], we need to apply several transformations. Let's outline each step in detail:
1. Vertical Transformation:
- The coefficient [tex]\( 0.35 \)[/tex] in front of the cosine function indicates a vertical compression. Since the amplitude of the parent function [tex]\( y = \cos(x) \)[/tex] is 1, multiplying by [tex]\( 0.35 \)[/tex] compresses it vertically by a factor of [tex]\( 0.35 \)[/tex].
2. Horizontal Period Change:
- The coefficient [tex]\( 8 \)[/tex] inside the cosine function affects the period. For the parent function [tex]\( y = \cos(x) \)[/tex], the period is [tex]\( 2\pi \)[/tex].
- The new period is given by dividing the original period by the horizontal compression factor:
[tex]\[
\text{New Period} = \frac{2\pi}{8} = \frac{\pi}{4}
\][/tex]
- Thus, there is a horizontal compression that changes the period to [tex]\( \frac{\pi}{4} \)[/tex].
3. Phase Shift:
- The term [tex]\( (x - \frac{\pi}{4}) \)[/tex] indicates a phase shift. Specifically, [tex]\( x \)[/tex] is shifted by [tex]\( \frac{\pi}{4} \)[/tex] units to the right.
Given the transformations outlined above, the correct sequence of transformations required to convert the parent function [tex]\( y = \cos(x) \)[/tex] to [tex]\( y = 0.35 \cos \left(8\left(x - \frac{\pi}{4}\right)\right) \)[/tex] are:
- Vertical compression by a factor of [tex]\( 0.35 \)[/tex]
- Horizontal compression to a period of [tex]\( \frac{\pi}{4} \)[/tex]
- Phase shift of [tex]\( \frac{\pi}{4} \)[/tex] units to the right
Therefore, the correct transformation selection matches the third option:
[tex]\[ \text{vertical compression of 0.35, horizontal compression to a period of} \ \frac{\pi}{4}, \text{phase shift of} \ \frac{\pi}{4} \ \text{units to the right} \][/tex]
