Question

vertical compression of \\(\frac{1}{2}\\), horizontal stretch to a period of \\(4\pi\\), vertical shift of 1 unit up, phase shift of \\(\pi\\) units left
vertical stretch of 2, horizontal compression to a period of \\(4\pi\\), vertical shift of 2 units up, phase shift of \\(\pi\\) units left
vertical stretch of \\(\frac{1}{2}\\), horizontal compression to a period of \\(2\pi\\), vertical shift of 1 unit up, phase shift of \\(\pi\\) units left
vertical compression of \\(\frac{1}{2}\\), horizontal stretch to a period of \\(2\pi\\), vertical shift of 1 unit down, phase shift of \\(\pi\\) units left

Explanation:

Step1: Analyze Vertical Transformation

The standard sine/cosine function has amplitude 1. The graph here has a maximum of 2? Wait, no, looking at the y - axis, the middle is around y = 1, peaks at y = 2? Wait, no, the grid: from y = - 1 to y = 2? Wait, no, the first option has vertical compression of 1/2. Let's think about the midline. The midline of the graph is at y = 1 (since it oscillates around y = 1, with minimums and maximums). The standard sine function \(y = \sin(x)\) has midline y = 0. So vertical shift is 1 unit up. The amplitude: the distance from midline to peak. If midline is 1, and peak is 2 (wait, no, the graph's peak is at y = 2? Wait, the y - axis has 1 and 2? Wait, the first option: vertical compression of 1/2. Let's assume the parent function is \(y=\sin(x)\) or \(y = \cos(x)\). Let's check the period. The period of the graph: from - 4π to 0, or from 0 to 4π? Wait, the x - axis has marks at - 4π, - 3π, - 2π, - π, 0, π, 2π, 3π, 4π. The distance between two consecutive peaks: from - 3π to π? Wait, no, looking at the graph, the period (distance between two identical points, like two peaks) is 4π. The period of \(y=\sin(x)\) is \(2\pi\). To get a period of \(4\pi\), we need a horizontal stretch by a factor of 2 (since period \(T=\frac{2\pi}{|B|}\), so \(4\pi=\frac{2\pi}{|B|}\Rightarrow |B|=\frac{1}{2}\), which is a horizontal stretch). Now vertical transformation: the amplitude. If the midline is 1, and the amplitude (distance from midline to peak) is 1 (since from y = 1 to y = 2? Wait, no, maybe the parent function is \(y=\cos(x)\) shifted. Wait, the first option: vertical compression of 1/2. Let's see: if we have a function \(y = A\cos(Bx - C)+D\). Midline \(D = 1\) (vertical shift up 1). Period \(T = 4\pi=\frac{2\pi}{|B|}\Rightarrow B=\frac{1}{2}\) (horizontal stretch). Amplitude \(|A|\): if the parent function has amplitude 1, and after compression by 1/2, amplitude is 1/2. Let's check the phase shift: phase shift is \(\frac{C}{B}\). The first option says phase shift of \(\pi\) units left, which means \(C=-B\pi\) (since phase shift left is \(C\) negative? Wait, phase shift formula: \(y = A\cos(B(x - h))+D\), phase shift is \(h\). So if phase shift is \(\pi\) left, \(h=-\pi\), so \(y = A\cos(B(x+\pi))+D\). Now let's check the options:

First option: vertical compression of 1/2 (A = 1/2), horizontal stretch to period 4π (so B = 1/2, period \(4\pi\)), vertical shift 1 unit up (D = 1), phase shift π units left (h = - π). This matches.

Let's check other options:

Second option: vertical stretch of 2 (A = 2), horizontal compression (period 4π would require B>1, but period \(T=\frac{2\pi}{|B|}\), so 4π=\frac{2\pi}{|B|}\Rightarrow B = 1/2, which is stretch, not compression), so second option is wrong.

Third option: horizontal compression (period 2π, which is same as parent, but our graph has period 4π), so wrong.

Fourth option: vertical shift 1 unit down (D=-1), but our midline is 1, so wrong.

Step2: Confirm Each Transformation

• Vertical Compression: The amplitude is reduced by 1/2, so vertical compression of 1/2.
• Horizontal Stretch: Period is 4π (twice the period of \(y = \cos(x)\) or \(y=\sin(x)\)), so horizontal stretch.
• Vertical Shift: Midline is at y = 1, so vertical shift 1 unit up.
• Phase Shift: The graph is shifted π units to the left.

Answer:

vertical compression of \(\frac{1}{2}\), horizontal stretch to a period of \(4\pi\), vertical shift of 1 unit up, phase shift of \(\pi\) units left