Question

the graph of $y = \cos x$ is transformed to the graph below. which statement about the graph is not correct? image of a cosine - like graph with options: a. the period is $\pi$; b. the amplitude is 2; c. the midline is $y = 1.5$; d. there is no phase shift

Explanation:

Step1: Recall cosine graph properties

The original \( y = \cos x \) has amplitude \( 1 \), period \( 2\pi \), midline \( y = 0 \), and no phase shift (starts at \( (0,1) \)).

Step2: Analyze the transformed graph

• Amplitude: The maximum value is \( 3 \)? Wait, no, looking at the graph, the peak is at \( y = 3 \)? Wait, no, the y-axis has marks. Wait, the original \( \cos x \) has max \( 1 \), min \( -1 \). The transformed graph: max is \( 3 \)? Wait, no, maybe the y-axis is labeled with \( 3 \) at top, \( -1 \) at bottom? Wait, no, let's re-examine. Wait, the midline: the midline is the average of max and min. If max is \( 3 \) and min is \( -1 \), midline is \( \frac{3 + (-1)}{2} = 1 \), not \( 1.5 \). Wait, option C says midline is \( y = 1.5 \). Let's check each option:
• Option A: Period. Original period \( 2\pi \), transformed: from \( 0 \) to \( \pi \), how many cycles? Wait, the graph shows two cycles in \( 2\pi \)? Wait, no, the x-axis: from \( 0 \) to \( 2\pi \), there are two full cycles? Wait, no, the original \( \cos x \) has period \( 2\pi \). The transformed graph: let's see the distance between peaks. If first peak at \( 0 \), next at \( \pi \), then period is \( \pi \). So A is correct.
• Option B: Amplitude. Amplitude is max - midline. If midline is \( 1 \) (wait, no, let's recalculate. Wait, the graph's max is \( 3 \), min is \( -1 \). Amplitude is \( 3 - 1 = 2 \) (since midline is \( 1 \), max - midline = \( 2 \)). So B is correct.
• Option C: Midline. Midline is \( \frac{\text{max} + \text{min}}{2} = \frac{3 + (-1)}{2} = 1 \), not \( 1.5 \). So C is incorrect.
• Option D: Phase shift. The graph starts at \( (0, 3) \), same as \( y = \cos x \) starts at \( (0,1) \) (scaled), so no phase shift. D is correct.

Answer:

C. The midline is \( y = 1.5 \)