Question

graph the function and state the amplitude, period, and midline. state the maximum al values and the corresponding x-values on one period for x ≥ 0.

f(x)=2 sin(1/2 x)

Explanation:

Step1: Find the amplitude

For a sine function \(y = A\sin(Bx)\), the amplitude is \(|A|\).
Here \(A = 2\), so the amplitude \(a=|2| = 2\).

Step2: Find the period

The period of the function \(y=\sin(Bx)\) is \(T=\frac{2\pi}{|B|}\).
Given \(B=\frac{1}{2}\), then \(T=\frac{2\pi}{\frac{1}{2}}=4\pi\).

Step3: Find the mid - line

For the function \(y = A\sin(Bx)\), the mid - line is \(y = 0\) (since there is no vertical shift \(D\) in the form \(y=A\sin(Bx - C)+D\)).

Step4: Find the maximum value and corresponding \(x\) - value

The maximum value of \(y=\sin t\) is \(1\). For \(y = 2\sin(\frac{1}{2}x)\), when \(\sin(\frac{1}{2}x)=1\).
We know that \(\sin t = 1\) when \(t=\frac{\pi}{2}+2k\pi,k\in\mathbb{Z}\). Let \(t=\frac{1}{2}x\), then \(\frac{1}{2}x=\frac{\pi}{2}+2k\pi\).
For \(k = 0\) (on the period \(x\geq0\)), \(x=\pi\) and \(y = 2\).

Answer:

• Amplitude: \(2\)
• Period: \(4\pi\)
• Mid - line: \(y = 0\)
• Maximum value: \(y = 2\) at \(x=\pi\)