Question

given the equation ( y = 4sin(2(x - 5)) + 7 )
the amplitude is:
the period is:
the horizontal shift is: units to the
the midline is: ( y = )

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: Identify the amplitude

For \(y = A\sin(B(x - C))+D\), the amplitude is \(|A|\). In \(y = 4\sin(2(x - 5))+7\), \(A = 4\), so the amplitude is \(|4|=4\).

Step3: Calculate the period

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

Step4: Determine the horizontal shift

The horizontal shift is \(C\). In \(y = 4\sin(2(x - 5))+7\), \(C = 5\), so the horizontal shift is \(5\) units to the right.

Step5: Find the midline

The midline of \(y = A\sin(B(x - C))+D\) is \(y = D\). Here \(D = 7\), so the midline is \(y = 7\).

Answer:

The amplitude is: \(4\)
The period is: \(\pi\)
The horizontal shift is: \(5\) units to the right
The midline is: \(y = 7\)