Question

graph the given function. state the period, amplitude, phase shift, and vertical shift of the function.
$y = - \sin \left( x + \frac { \pi } { 4 } \
ight)$
period:
amplitude:
phase shift:
vertical shift:
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Explanation:

Step1: Recall the general form of sine function

The general form of a sine function is \(y = A\sin(B(x - C))+D\). For the given function \(y=-\sin(x + \frac{\pi}{4})\), we can rewrite it as \(y=-1\times\sin(1\times(x-(-\frac{\pi}{4})))+0\).

Step2: Calculate the amplitude

The amplitude \(|A|\). Here \(A=-1\), so \(|A| = 1\).

Step3: Calculate the period

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 phase - shift

The phase - shift is \(C\). Here \(C=-\frac{\pi}{4}\), so the phase - shift is \(\frac{\pi}{4}\) units to the left.

Step5: Calculate the vertical - shift

The vertical - shift is \(D\). Here \(D = 0\).

Answer:

Period: \(2\pi\)
Amplitude: \(1\)
Phase Shift: \(\frac{\pi}{4}\) units to the left
Vertical Shift: \(0\)