Question

y = sin(x + \frac{\pi}{2})
graph:
phase shift:
vertical shift:
period:
amplitude:

Explanation:

Step1: Recall the general form of sine function

The general form of a sine function is \(y = A\sin(B(x - C))+D\), where \(A\) is the amplitude, \(B\) affects the period (\(T=\frac{2\pi}{|B|}\)), \(C\) is the phase - shift (shift to the right if \(C>0\) and to the left if \(C < 0\)), and \(D\) is the vertical shift.
For the function \(y=\sin(x+\frac{\pi}{2})\), we can rewrite it as \(y=\sin(1\times(x-(-\frac{\pi}{2})))+0\).

Step2: Calculate the amplitude

For \(y = A\sin(B(x - C))+D\), comparing \(y=\sin(x+\frac{\pi}{2})\) with \(y = A\sin(B(x - C))+D\), we have \(A = 1\). So the amplitude \(|A|=1\).

Step3: Calculate the period

Since \(B = 1\), using the formula \(T=\frac{2\pi}{|B|}\), we get \(T=\frac{2\pi}{|1|}=2\pi\).

Step4: Calculate the phase - shift

Since \(C=-\frac{\pi}{2}\), the phase - shift is \(-\frac{\pi}{2}\) (a shift to the left by \(\frac{\pi}{2}\) units).

Step5: Calculate the vertical shift

Since \(D = 0\), the vertical shift is \(0\).

Step6: Graph description

The graph of \(y=\sin(x)\) is shifted to the left by \(\frac{\pi}{2}\) units. We know that \(\sin(x+\frac{\pi}{2})=\cos(x)\). The graph of \(y = \cos(x)\) is a cosine curve with amplitude \(1\), period \(2\pi\), no vertical shift, and a phase - shift (equivalent to the standard cosine function \(y=\cos(x)\) which can be thought of as a left - shifted sine function by \(\frac{\pi}{2}\) units)

Answer:

• Graph: The graph of \(y = \sin(x)\) shifted left by \(\frac{\pi}{2}\) units (or the graph of \(y=\cos(x)\))
• Phase shift: \(-\frac{\pi}{2}\) (left - shifted by \(\frac{\pi}{2}\) units)
• Vertical shift: \(0\)
• Period: \(2\pi\)
• Amplitude: \(1\)