Question

periodic functions periodic functions unit test
write the equation of the periodic function based on the graph.
(1 point)
\\(\bigcirc\\ y = -3\cos x\\)
\\(\bigcirc\\ y = -3\sin x\\)
\\(\bigcirc\\ y = -\frac{1}{3}\cos x\\)
\\(\bigcirc\\ y = 3\cos x\\)

Explanation:

Step1: Analyze the graph's shape and key points

The graph is a cosine - like curve (starts at a maximum or minimum on the y - axis). The standard cosine function \(y = \cos x\) has a maximum of 1 at \(x = 0\) and a minimum of - 1 at \(x=\pi\). The given graph has a minimum at \(x = 0\) with \(y=-3\).

Step2: Analyze the amplitude and sign

The general form of a cosine function is \(y = A\cos x\), where \(|A|\) is the amplitude. The amplitude here is 3 (since the distance from the mid - line to the maximum or minimum is 3). The negative sign in front of \(A\) reflects the graph over the x - axis. For \(y=-3\cos x\), when \(x = 0\), \(y=-3\cos(0)=-3\times1=-3\), which matches the y - intercept of the graph.
For \(y = - 3\sin x\), when \(x = 0\), \(y=-3\sin(0)=0\), which does not match the y - intercept (\(y = - 3\) at \(x = 0\)) of the given graph.
For \(y=-\frac{1}{3}\cos x\), when \(x = 0\), \(y =-\frac{1}{3}\cos(0)=-\frac{1}{3}\), which does not match the y - intercept of the graph.
For \(y = 3\cos x\), when \(x = 0\), \(y=3\cos(0)=3\), which does not match the y - intercept (\(y=-3\)) of the graph.

Answer:

\(y=-3\cos x\) (the option: \(y = - 3\cos x\))