Question

question 6
given the equation ( y = 7 sin left( \frac { pi } { 3 } x + \frac { 4 pi } { 3 }
ight) + 2 )
the amplitude is:
the period is:
the horizontal shift is: units to the select an answer
the midline is: ( y = )
question help: video

Explanation:

Step1: Find the amplitude

For the sine function \(y = A\sin(Bx - C)+D\), the amplitude is \(|A|\).
In the given function \(y = 7\sin(\frac{\pi}{3}x+\frac{4\pi}{3}) + 2\), \(A = 7\), so the amplitude is \(|7|=7\).

Step2: Find the period

The period of the sine function \(y = A\sin(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\).
Here \(B=\frac{\pi}{3}\), so \(T=\frac{2\pi}{\frac{\pi}{3}}=6\).

Step3: Find the horizontal shift

Rewrite the function in the form \(y = A\sin(B(x - h))+D\).
\(y = 7\sin(\frac{\pi}{3}(x + 4))+2\), so the horizontal shift \(h=- 4\), which is \(4\) units to the left.

Step4: Find the mid - line

For the sine function \(y = A\sin(Bx - C)+D\), the mid - line is \(y = D\).
Here \(D = 2\), so the mid - line is \(y = 2\).

Answer:

The amplitude is: \(7\)
The period is: \(6\)
The horizontal shift is: \(4\) units to the left
The midline is: \(y = 2\)