Question

more practice graphing vertical and horizontal translations

1. graph one complete primary period of the function ( y = 3 cos left( \frac { pi } { 3 } t - \frac { 2 pi } { 3 }

ight) + 5 ) and provide the requested information (use interval notation as appropriate).

amplitude:
vertical shift:
average value:
horizontal shift:
maximum value:
endpoints of the primary interval:
minimum value:
domain:
period:
range:

Explanation:

Step1: Find the endpoints of the primary interval

For the cosine function \(y = A\cos(B(t - C))+D\), the primary interval of \(y=\cos t\) is \([0, 2\pi]\). For \(y = 3\cos(\frac{\pi}{3}(t - 2))+5\), we set \(\frac{\pi}{3}(t - 2)=0\) and \(\frac{\pi}{3}(t - 2)=2\pi\).

• When \(\frac{\pi}{3}(t - 2)=0\), solve for \(t\):

\(\frac{\pi}{3}(t - 2)=0\Rightarrow t-2 = 0\Rightarrow t = 2\)

• When \(\frac{\pi}{3}(t - 2)=2\pi\), solve for \(t\):

\(\frac{\pi}{3}(t - 2)=2\pi\Rightarrow t-2=6\Rightarrow t = 8\)
So the endpoints of the primary interval are \(t = 2\) and \(t = 8\), and the interval is \([2,8]\)

Step2: Find the domain

Since the cosine function is periodic with period \(P=\frac{2\pi}{B}\), and \(B=\frac{\pi}{3}\), \(P = 6\). The domain of a cosine function \(y = 3\cos(\frac{\pi}{3}(t - 2))+5\) is all real numbers, which in interval notation is \((-\infty,\infty)\)

Step3: Find the range

The amplitude \(A = 3\) and the vertical - shift \(D = 5\). The range of \(y=\cos x\) is \([- 1,1]\). For \(y = 3\cos(\frac{\pi}{3}(t - 2))+5\), we use the formula \(y=A\cos(\cdots)+D\)

• The maximum value: \(y_{\text{max}}=A + D=3 + 5=8\)
• The minimum value: \(y_{\text{min}}=-A + D=-3 + 5 = 2\)

So the range is \([2,8]\)

Answer:

Endpoints of the Primary Interval: \([2,8]\)
Domain: \((-\infty,\infty)\)
Range: \([2,8]\)