Question

which transformations are needed to change the parent cosine function to y = 3\cos\left(10(x - \pi)\
ight)?
\bigcirc vertical compression of 3, horizontal stretch to a period of 5\pi, phase shift of \pi units to the left
\bigcirc vertical stretch of 3, horizontal compression to a period of \frac{\pi}{5}, phase shift of \pi units to the right
\bigcirc vertical compression of 3, horizontal stretch to a period of 10p, phase shift of \pi units to the right
\bigcirc vertical stretch of 3, horizontal compression to a period of \frac{\pi}{5}, phase shift of \pi units to the left

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine function is \( y = A\cos(B(x - C)) + D \), where:

• \( |A| \) is the vertical stretch/compression factor (stretch if \( |A|>1 \), compression if \( 0<|A|<1 \)),
• The period is \( \frac{2\pi}{|B|} \),
• \( C \) is the phase shift (right if \( C>0 \), left if \( C<0 \)).

Step2: Analyze \( A \) in \( y = 3\cos(10(x - \pi)) \)

Here, \( A = 3 \). Since \( |3|>1 \), this is a vertical stretch by a factor of 3.

Step3: Analyze \( B \) to find the period

For the given function, \( B = 10 \). The period of a cosine function is \( \frac{2\pi}{|B|} \). Substituting \( B = 10 \), we get the period as \( \frac{2\pi}{10}=\frac{\pi}{5} \). Since \( |B| = 10>1 \), the graph is horizontally compressed (because a larger \( B \) value makes the period shorter, compressing the graph horizontally).

Step4: Analyze \( C \) to find the phase shift

In the general form \( y = A\cos(B(x - C)) + D \), the phase shift is \( C \) units. Here, \( C=\pi \), so the phase shift is \( \pi \) units to the right (because \( C>0 \)).

Answer:

B. vertical stretch of 3, horizontal compression to a period of \(\frac{\pi}{5}\), phase shift of \(\pi\) units to the right