Question

which transformations are needed to change the parent sine function to the sine function below? (image of a sine graph with x-axis labeled -4π, -3π, -2π, -π, π, 2π, 3π, 4π and y-axis labeled -1, 1, 2, 3) options: 1. vertical compression of 1/2, horizontal stretch to a period of 4π, vertical shift of 1 unit up, phase shift of π units left 2. vertical stretch of 2, horizontal compression to a period of 4π, vertical shift of 2 units up, phase shift of π units left

Explanation:

Step1: Analyze Vertical Transformation

The parent sine function \( y = \sin(x) \) has an amplitude of 1 (range \([-1, 1]\)). The given graph has a range from \( 0 \) to \( 2 \) (midline at \( y = 1 \), amplitude \( 1 \)). Wait, no—wait, the midline: parent sine has midline \( y = 0 \). The given graph's midline is \( y = 1 \) (since the average of the maximum (1.5? Wait, no, looking at the graph: the peaks are at \( y = 1.5 \)? Wait, no, the grid: the y-axis has 1, 2, 3. Wait, the graph's maximum is at \( y = 1.5 \)? No, wait the first option says vertical compression of \( \frac{1}{2} \), but let's check period.

Step2: Analyze Horizontal Transformation (Period)

Parent sine function \( y = \sin(x) \) has period \( 2\pi \). The given graph: from \( -2\pi \) to \( 2\pi \) is one period? Wait, no, looking at the x-axis: the graph repeats every \( 4\pi \)? Wait, from \( -4\pi \) to \( 0 \), and \( 0 \) to \( 4\pi \)? Wait, no, the peaks: at \( x = 0 \), a peak, then next peak at \( x = 4\pi \)? No, wait the x-axis labels: \( -4\pi, -3\pi, -2\pi, -\pi, 0, \pi, 2\pi, 3\pi, 4\pi \). The graph has a peak at \( x = 0 \), then a trough at \( x = 2\pi \), then peak at \( x = 4\pi \)? Wait, no, the distance between two consecutive peaks: from \( x = -4\pi \) peak to \( x = 0 \) peak is \( 4\pi \), so period is \( 4\pi \). Parent period is \( 2\pi \), so to get period \( 4\pi \), we do a horizontal stretch by factor \( 2 \) (since period \( T = \frac{2\pi}{b} \), so \( 4\pi = \frac{2\pi}{b} \implies b = \frac{1}{2} \), which is a horizontal stretch by 2, or period \( 4\pi \) is a horizontal stretch).

Step3: Analyze Vertical Shift

Parent sine has midline \( y = 0 \). The given graph's midline is \( y = 1 \) (since the average of the minimum and maximum: the minimum is at \( y = 0 \)? Wait, no, the graph's minimum is at \( y = 0 \)? Wait, the y-axis has 0 (between -1 and 1? Wait, the grid: the bottom is -1, then 1, 2, 3. Wait, the graph's minimum is at \( y = 0 \) (since the troughs are at \( y = 0 \)? No, the first option says vertical shift 1 unit up. Let's check the first option: vertical compression of \( \frac{1}{2} \) (amplitude becomes \( 1 \times \frac{1}{2} = 0.5 \)), horizontal stretch to period \( 4\pi \) (correct, as period is \( 4\pi \)), vertical shift 1 unit up (midline from 0 to 1), phase shift \( \pi \) units left? Wait, parent sine at \( x = 0 \) is 0, but the given graph at \( x = 0 \) is a peak. Wait, parent sine \( y = \sin(x) \) has a peak at \( x = \frac{\pi}{2} \), but here at \( x = 0 \) is a peak. So phase shift: if we shift \( \pi \) units left, \( y = \sin(x + \pi) = -\sin(x) \), but with vertical shift and compression. Wait, maybe the first option: vertical compression (amplitude 0.5), horizontal stretch (period 4π), vertical shift 1 up, phase shift π left. Let's check the second option: vertical stretch of 2 (amplitude 2), but the graph's amplitude is 1 (since midline 1, max 2, min 0: amplitude 1), so vertical stretch of 2 would make amplitude 2, which is wrong. So first option is correct.

Answer:

A. 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 (assuming the first option is A, as per the options given)