Question

ii. graph each function by finding the amplitude, period, phase shift, and vertical shift.
9.) ( y = 2 cos x - 3 )
amp = _ pd = _
shift = _ v. shift = _
10.) ( y = sin ( x + pi ) + 4 )
amp = _ pd = _
p. shift = _ v. shift = _
11.) ( y = \tan left( \frac { 1 } { 2 } x
ight) + 1 )
amp = _ pd = _
p. shift = _ v. shift = _

Explanation:

Step1: Amplitude formula

For \(y = A\cos(x)+k\), amplitude is \(|A|\). For \(y = 2\cos(x)-3\), \(A = 2\), so amplitude \(=|2| = 2\).

Step2: Period formula

For \(y = A\cos(Bx)+k\), period is \(\frac{2\pi}{|B|}\). Here \(B = 1\), so period \(=\frac{2\pi}{|1|}=2\pi\).

Step3: Phase - shift formula

For \(y = A\cos(B(x - C))+k\), phase - shift is \(C\). For \(y = 2\cos(x)-3\) (equivalent to \(y = 2\cos(1(x - 0))-3\)), phase - shift \(=0\) (none).

Step4: Vertical - shift formula

For \(y = A\cos(x)+k\), vertical - shift is \(k\). Here \(k=-3\), so vertical - shift is down \(3\).

Answer:

Amplitude \( = 2\), Period \(=2\pi\), Phase - shift \(=\text{none}\), Vertical - shift \(=\text{down }3\)