Question

the graph of $y = \cos x$ is transformed to the graph below. which statement about the graph is not correct? a. the phase shift is $\frac{\pi}{2}$. b. the midline is $y = 1$. c. the period is $\pi$. d. the amplitude is 2.

Explanation:

Step1: Analyze Phase Shift

The original \( y = \cos x \) has a maximum at \( x = 0 \). The transformed graph also has a maximum at \( x = 0 \), so phase shift is \( 0 \), not \( \frac{\pi}{2} \).

Step2: Analyze Midline

Midline is the average of max and min. Max is \( 3 \), min is \( -1 \). Midline: \( \frac{3 + (-1)}{2} = 1 \), so \( y = 1 \) is correct.

Step3: Analyze Period

Original period of \( \cos x \) is \( 2\pi \). The transformed graph completes 2 cycles in \( 2\pi \), so period is \( \pi \), correct.

Step4: Analyze Amplitude

Amplitude is distance from midline to max. Midline \( y = 1 \), max \( y = 3 \), so amplitude \( 3 - 1 = 2 \), correct.

Answer:

A. The phase shift is \(\frac{\pi}{2}\)