Question

day 2 graphing sine and cosine worksheet (graded)

1. find all possible angles for θ using the domain ( 0 leq \theta < 2pi ).

a) ( sin \theta = \frac{sqrt{3}}{2} ) b) ( cos \theta = -\frac{1}{2} ) c) ( cos \theta = 1 ) d) ( cos \theta = \frac{sqrt{2}}{2} )

1. ( y = 4 sin ( 2 x ) )

phase shift: ______ period: ______
vertical shift: ______ amplitude: ______

1. ( y = 2 sin ( 4 x ) )

phase shift: ______ period: ______
vertical shift: ______ amplitude: ______

1. ( y = 2 + cos ( x - \frac { pi } { 4 } ) )

phase shift: ______ period: ______
vertical shift: ______ amplitude: ______

1. ( y = - 3 - cos ( x ) )

phase shift: ______ period: ______
vertical shift: ______ amplitude: ______

Explanation:

Step1: Recall the general form of a sinusoidal function

The general form of a sinusoidal function is \(y = A\sin(B(x - C))+D\) or \(y = A\cos(B(x - C))+D\), where:

• Amplitude \(|A|\)
• Period \(T=\frac{2\pi}{|B|}\)
• Phase - shift \(C\) (positive is to the right, negative is to the left)
• Vertical - shift \(D\)

Step2: Solve for problem 2 (\(y = 4\sin(2x)\))

• For \(y = 4\sin(2x)\), comparing with \(y = A\sin(B(x - C))+D\), we have \(A = 4\), \(B = 2\), \(C = 0\), \(D = 0\)
• Amplitude: \(|A|=4\)
• Period: \(T=\frac{2\pi}{|B|}=\frac{2\pi}{2}=\pi\)
• Phase - shift: \(C = 0\)
• Vertical - shift: \(D = 0\)

Step3: Solve for problem 3 (\(y = 2\sin(4x)\))

• For \(y = 2\sin(4x)\), comparing with \(y = A\sin(B(x - C))+D\), we have \(A = 2\), \(B = 4\), \(C = 0\), \(D = 0\)
• Amplitude: \(|A|=2\)
• Period: \(T=\frac{2\pi}{|B|}=\frac{2\pi}{4}=\frac{\pi}{2}\)
• Phase - shift: \(C = 0\)
• Vertical - shift: \(D = 0\)

Step4: Solve for problem 4 (\(y = 2+\cos(x-\frac{\pi}{4})\))

• For \(y = 2+\cos(x - \frac{\pi}{4})\), comparing with \(y=A\cos(B(x - C))+D\), we have \(A = 1\), \(B = 1\), \(C=\frac{\pi}{4}\), \(D = 2\)
• Amplitude: \(|A| = 1\)
• Period: \(T=\frac{2\pi}{|B|}=2\pi\)
• Phase - shift: \(C=\frac{\pi}{4}\) (shift to the right by \(\frac{\pi}{4}\))
• Vertical - shift: \(D = 2\)

Step5: Solve for problem 5 (\(y=-3-\cos(x)\))

• For \(y=-3-\cos(x)\), comparing with \(y = A\cos(B(x - C))+D\), we have \(A=- 1\), \(B = 1\), \(C = 0\), \(D=-3\)
• Amplitude: \(|A| = 1\)
• Period: \(T=\frac{2\pi}{|B|}=2\pi\)
• Phase - shift: \(C = 0\)
• Vertical - shift: \(D=-3\)

Answer:

1. Phase shift: \(0\); Period: \(\pi\); Vertical shift: \(0\); Amplitude: \(4\)
2. Phase shift: \(0\); Period: \(\frac{\pi}{2}\); Vertical shift: \(0\); Amplitude: \(2\)
3. Phase shift: \(\frac{\pi}{4}\); Period: \(2\pi\); Vertical shift: \(2\); Amplitude: \(1\)
4. Phase shift: \(0\); Period: \(2\pi\); Vertical shift: \(-3\); Amplitude: \(1\)