Question

determine the amplitude, period, and phase shift of the function. then graph the function.
$y = \frac{1}{3} \cos (2x - \pi)$
the amplitude is $\frac{1}{3}$.
the period is $\pi$.
(type an exact answer, using $\pi$ as needed. use integers or fractions for any numbers in the expression.)
the phase shift is
(type an exact answer, using $\pi$ as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the general form of cosine function

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

• Amplitude \( = |A| \)
• Period \( = \frac{2\pi}{|B|} \)
• Phase shift \( = \frac{C}{B} \)
• Vertical shift \( = D \)

For the given function \( y = \frac{1}{3}\cos(2x - \pi) \), we can identify \( A = \frac{1}{3} \), \( B = 2 \), \( C = \pi \), and \( D = 0 \).

Step2: Calculate the phase shift

Using the formula for phase shift \( \frac{C}{B} \), substitute \( C = \pi \) and \( B = 2 \):

$$ \text{Phase shift} = \frac{\pi}{2} $$

Answer:

\(\frac{\pi}{2}\)