1.
Determine the horizontal translation of the function \(y=5\sin\left(\frac12x-6\right)\).
Solution: \(y=5\sin(\frac12(x-12))\). Horizontal translation is 12 units right. Answer: A
2.
For \(f(x)=2\cos\left(8x+\frac{\pi}{3}\right)+1\), determine the period and phase shift.
Solution: Period \(2\pi/8=\pi/4\); phase shift \(\pi/3\) left. Answer: C
3.
For \(f(x)=-\sin\!\left(3\left(x-\frac{\pi}{6}\right)\right)+5\), determine the period and phase shift.
Solution: Period \(2\pi/3\); shift right \(\pi/6\). Answer: B
4.
The graph of a periodic trigonometric function of the form y=a sin(x-c)+d is shown. Determine d.
Solution: d is the equilibrium line: (max+min)/2=-1. Answer: B
5.
At a seaport, \(d=3.4\sin\!\left(2\pi\frac{t-7}{10.6}\right)+2.8\). Determine the depth at 6:30 PM.
Solution: Use \(t=18.5\). Substitution gives \(d\approx4.53\) m. Answer: C
6.
Determine all values of \(x\), \(0\le x<\pi\), for which \(\tan(2x)\) is undefined.
Solution: \(\tan\theta\) undefined at \(\theta=\pi/2+k\pi\). Hence \(2x=\pi/2,3\pi/2\), so \(x=\pi/4,3\pi/4\). Answer: B
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