To determine the transformations needed to change the parent cosine function [tex]\( y = \cos(x) \)[/tex] to [tex]\( y = 3 \cos(10(x - \pi)) \)[/tex], we need to break down the transformations step by step:
1. Vertical Stretch/Compression:
- The parent function [tex]\( y = \cos(x) \)[/tex] is modified to [tex]\( y = 3 \cos(x) \)[/tex].
- The coefficient 3 in front of the cosine function indicates a vertical stretch by a factor of 3.
2. Horizontal Compression/Stretch:
- The argument inside the cosine function [tex]\( x \)[/tex] is modified to [tex]\( 10(x - \pi) \)[/tex].
- The coefficient 10 indicates a horizontal compression. The period of the parent cosine function is [tex]\( 2\pi \)[/tex]. When we multiply [tex]\( x \)[/tex] by 10, the period becomes:
[tex]\[
\text{New Period} = \frac{2\pi}{10} = \frac{\pi}{5}
\][/tex]
3. Phase Shift:
- The argument [tex]\( x \)[/tex] is modified to [tex]\( x - \pi \)[/tex].
- The subtraction of [tex]\( \pi \)[/tex] indicates a phase shift. Specifically, it shifts the graph [tex]\( \pi \)[/tex] units to the right.
Given these transformations, the correct option is:
- Vertical stretch of 3
- Horizontal compression to a period of [tex]\(\frac{\pi}{5}\)[/tex]
- Phase shift of [tex]\(\pi\)[/tex] units to the right
Therefore, the correct answer is:
Vertical stretch of 3, horizontal compression to a period of [tex]\( \frac{\pi}{5} \)[/tex], phase shift of [tex]\( \pi \)[/tex] units to the right
This matches choice number two:
- Vertical stretch of 3 , horizontal compression to a period of [tex]\( \frac{\pi}{5} \)[/tex] , phase shift of [tex]\( \pi \)[/tex] units to the right
