Question

assignment 8.1: graphs of the sine and cosine functions
score: 5/21 answered: 3/10
question 4
given the equation ( y = 7 sin ( 4 ( x + 6 ) ) + 3 )
the amplitude is:
the period (exact answer) is:
the horizontal shift is: units to the
the midline is: ( y = )
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Explanation:

Step1: Find the amplitude

For a sine function \(y = A\sin(B(x - C))+D\), the amplitude is \(|A|\).
In \(y = 7\sin(4(x + 6))+3\), \(A = 7\), so the amplitude is \(|7|=7\).

Step2: Find the period

The period of \(y = A\sin(B(x - C))+D\) is \(\frac{2\pi}{|B|}\).
Here \(B = 4\), so the period is \(\frac{2\pi}{4}=\frac{\pi}{2}\).

Step3: Find the horizontal shift

For \(y = A\sin(B(x - C))+D\), the horizontal shift is \(C\).
The equation is \(y = 7\sin(4(x-(- 6)))+3\), so the horizontal shift is \(6\) units to the left.

Step4: Find the mid - line

For \(y = A\sin(B(x - C))+D\), the mid - line is \(y = D\).
Here \(D = 3\), so the mid - line is \(y = 3\).

Answer:

The amplitude is \(7\).
The period is \(\frac{\pi}{2}\).
The horizontal shift is \(6\) units to the left.
The midline is \(y = 3\).