Question

using the graph shown below, determine the function the graph depicts.
a. $y = \cos 2x$
b. $y = 2\sin x$
c. $y = \sin 2x$
d. $y = 2\cos x$

Explanation:

Step1: Recall the general form of sine function

The general form of a sine function is \(y = A\sin(Bx)\), where \(A\) is the amplitude and \(B\) affects the period (\(T=\frac{2\pi}{B}\)).

Step2: Analyze the amplitude

The amplitude is the maximum value of the function. From the graph, the maximum value is \(2\) and the minimum is \(- 2\), so \(A = 2\).

Step3: Analyze the period

The period of the standard sine function \(y=\sin(x)\) is \(2\pi\). For the given graph, when \(x = \pi\), the function completes one full cycle (from \(0\) to \(0\) passing through maximum and minimum). Using the period formula \(T=\frac{2\pi}{B}\), with \(T=\pi\), we get \(\pi=\frac{2\pi}{B}\), so \(B = 2\).

Answer:

B. \(y = 2\sin x\)