Question

state the period, amplitude, phase shift, and vertical shift of the function.
$y = -\sin\left(x + \frac{\pi}{2}\
ight)$
period:
amplitude:
phase shift:
vertical shift:
question help: video ebook message instructor

Explanation:

Step1: Recall the general form of sine function

The general form of a sine function is \(y = A\sin(B(x - C))+D\).

Step2: Identify the values of \(A\), \(B\), \(C\), \(D\)

For the function \(y =-\sin(x+\frac{\pi}{2})\), we have \(A=- 1\), \(B = 1\), \(C=-\frac{\pi}{2}\), \(D = 0\).

Step3: Calculate the period

The formula for the period of \(y = A\sin(B(x - C))+D\) is \(T=\frac{2\pi}{|B|}\). Since \(B = 1\), \(T=\frac{2\pi}{|1|}=2\pi\).

Step4: Calculate the amplitude

The formula for the amplitude is \(|A|\). Since \(A=-1\), \(|A| = 1\).

Step5: Calculate the phase - shift

The formula for the phase - shift is \(C\). Since \(C =-\frac{\pi}{2}\), the phase - shift is \(-\frac{\pi}{2}\) (or \(\frac{\pi}{2}\) to the left).

Step6: Calculate the vertical - shift

The formula for the vertical - shift is \(D\). Since \(D = 0\), the vertical - shift is \(0\).

Answer:

Period: \(2\pi\)
Amplitude: \(1\)
Phase Shift: \(-\frac{\pi}{2}\)
Vertical Shift: \(0\)