Question

1 multiple answer 1 point
select all the correct statements
□ the amplitude of ( y = 2 cos ( \frac { 1 } { 2 } x ) ) is ( \frac { 1 } { 2 } )
□ the period of ( y = 2 cos ( \frac { 1 } { 2 } x ) ) is ( prod )
□ the amplitude of ( y = \frac { prod } { 2 } sin ( \frac { prod } { 2 } x ) ) is ( \frac { prod } { 2 } )
□ the period of ( y = 4 prod sin ( \frac { 1 } { 2 } x ) ) is ( 4 prod )
□ the period of ( y = cos ( 2 prod x ) ) is 1
□ the amplitude of ( y = 2 sin ( \frac { 1 } { 2 } x ) ) is 2
2 multiple choice 1 point
the equation for f and the graph of g are given. how do the period and the amplitude of the functions compare?
( f ( x ) = 2 cos ( \frac { prod } { 2 } x ) )
the amplitudes are the same, but the period of f is half as long as the period of g.
the amplitudes are the same, but the period of f is twice as long as the period of g.
the periods are the same, but the amplitude of f is half as great as the amplitude of g.
the periods are the same, but the amplitude of f is twice as great as the amplitude of g

Explanation:

Question 1

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: Analyze \(y = 2\cos(\frac{1}{2}x)\)

• Amplitude: \(|A| = 2

eq\frac{1}{2}\)

• Period: \(T=\frac{2\pi}{\frac{1}{2}}=4\pi

eq\pi\)

Step3: Analyze \(y=\frac{\pi}{2}\sin(\frac{\pi}{2}x)\)

• Amplitude: \(|A|=\frac{\pi}{2}\)
• Period: \(T = \frac{2\pi}{\frac{\pi}{2}}=4\)

Step4: Analyze \(y = 4\pi\sin(\frac{1}{2}x)\)

• Amplitude: \(|A| = 4\pi\)
• Period: \(T=\frac{2\pi}{\frac{1}{2}}=4\pi\)

Step5: Analyze \(y=\cos(2\pi x)\)

• Amplitude: \(|A| = 1\)
• Period: \(T=\frac{2\pi}{2\pi}=1\)

Step6: Analyze \(y = 2\sin(\frac{1}{2}x)\)

• Amplitude: \(|A|=2\)
• Period: \(T=\frac{2\pi}{\frac{1}{2}}=4\pi\)

Step1: Find the amplitude and period of \(f(x)=2\cos(\frac{\pi}{2}x)\)

• Amplitude: \(|A| = 2\)
• Period: \(T_f=\frac{2\pi}{\frac{\pi}{2}}=4\)

Assume \(g(x)=\sin(x)\) (from the general sine - wave graph shape, amplitude \(A = 2\) for \(g\) (since the vertical range from - 2 to 2), and period \(T_g = 2\) (counting the number of cycles in a given \(x\) - range).

Step2: Compare amplitude and period

• Amplitude: \(|A_f|=|A_g| = 2\)
• Period: \(T_f=\frac{2\pi}{\frac{\pi}{2}}=4\), if \(T_g = 2\), \(T_f = 2T_g\)

Answer:

The amplitude of \(y=\frac{\pi}{2}\sin(\frac{\pi}{2}x)\) is \(\frac{\pi}{2}\), the period of \(y = 4\pi\sin(\frac{1}{2}x)\) is \(4\pi\), the period of \(y=\cos(2\pi x)\) is \(1\), the amplitude of \(y = 2\sin(\frac{1}{2}x)\) is \(2\)

Question 2