Question

identify each characteristic of the trig function, then graph.
4.
$y = -2\cos\left(x + \frac{\pi}{4}\
ight)$
amplitude
period
phase shift
vertical shift
midline

Explanation:

Step1: Find the amplitude

For a cosine function \(y = A\cos(B(x - C))+D\), the amplitude is \(|A|\).
Here \(A=- 2\), so the amplitude is \(| - 2|=2\).

Step2: Find the period

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

Step3: Find the phase - shift

The phase - shift of the function \(y = A\cos(B(x - C))+D\) is \(C\).
The function is \(y=-2\cos(x+\frac{\pi}{4})=-2\cos(1\times(x-(-\frac{\pi}{4})))+0\), so the phase - shift is \(-\frac{\pi}{4}\) (shift to the left \(\frac{\pi}{4}\) units).

Step4: Find the vertical shift

For the function \(y = A\cos(B(x - C))+D\), the vertical shift is \(D\).
Here \(D = 0\), so the vertical shift is \(0\).

Step5: Find the midline

The midline of the function \(y = A\cos(B(x - C))+D\) is \(y = D\).
Since \(D = 0\), the midline is \(y = 0\).

Answer:

Amplitude: \(2\); Period: \(2\pi\); Phase Shift: \(-\frac{\pi}{4}\); Vertical Shift: \(0\); Midline: \(y = 0\)