Question

name:
date:
mc precalculus: section 8.2 & 8.3 practice

1. give the period and amplitude of each

a. ( y = 4cos(2x) )
b. ( y = - 4sinleft(\frac{x}{3}
ight) )
c. ( y = 5sinleft(\frac{2pi}{7}x
ight) )
d. ( y = 1.5cosleft(\frac{pi}{3}x
ight) )

Explanation:

Step1: Recall the formula for amplitude and period

For a function \(y = A\sin(Bx)\) or \(y=A\cos(Bx)\), the amplitude is \(|A|\) and the period is \(T=\frac{2\pi}{|B|}\).

Step2: Solve for part a

For \(y = 4\cos(2x)\), \(A = 4\), \(B = 2\).
Amplitude: \(|4|=4\).
Period: \(T=\frac{2\pi}{2}=\pi\).

Step3: Solve for part b

For \(y=-4\sin(\frac{x}{3})\), \(A=-4\), \(B=\frac{1}{3}\).
Amplitude: \(|-4| = 4\).
Period: \(T=\frac{2\pi}{\frac{1}{3}}=6\pi\).

Step4: Solve for part c

For \(y = 5\sin(\frac{2\pi}{7}x)\), \(A = 5\), \(B=\frac{2\pi}{7}\).
Amplitude: \(|5|=5\).
Period: \(T=\frac{2\pi}{\frac{2\pi}{7}}=7\).

Step5: Solve for part d

For \(y = 1.5\cos(\frac{\pi}{3}x)\), \(A = 1.5\), \(B=\frac{\pi}{3}\).
Amplitude: \(|1.5|=1.5\).
Period: \(T=\frac{2\pi}{\frac{\pi}{3}}=6\).

Answer:

a. Amplitude: \(4\), Period: \(\pi\)
b. Amplitude: \(4\), Period: \(6\pi\)
c. Amplitude: \(5\), Period: \(7\)
d. Amplitude: \(1.5\), Period: \(6\)