Question

chapter 6 quiz
170 points possible answered: 11/17
question 12
given the equation ( y = 4 sin ( 7 ( x + 8 ) ) + 3 )
the amplitude is:
the period is:
the horizontal shift is: units to the
the midline is: ( y = )
add work

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Explanation:

Step1: Recall the general form of a sine function

The general form of a sine function is \(y = A\sin(B(x - C))+D\).

Step2: Find the amplitude

For the function \(y = 4\sin(7(x + 8))+3\), comparing with \(y = A\sin(B(x - C))+D\), \(A = 4\). The amplitude is \(|A|\), so the amplitude is \(4\).

Step3: Find the period

The formula for the period of a sine function \(y = A\sin(B(x - C))+D\) is \(T=\frac{2\pi}{|B|}\). Here \(B = 7\), so \(T=\frac{2\pi}{7}\).

Step4: Find the horizontal shift

The formula for the horizontal shift is \(C\). The function is \(y = 4\sin(7(x+8)) + 3=4\sin(7(x-(- 8)))+3\), so \(C=-8\). The horizontal shift is \(8\) units to the left.

Step5: Find the mid - line

The formula for the mid - line is \(y = D\). Here \(D = 3\), so the mid - line is \(y = 3\).

Answer:

The amplitude is \(4\).
The period is \(\frac{2\pi}{7}\).
The horizontal shift is \(8\) units to the left.
The midline is \(y = 3\).