Question

enter the equation for the graph. y = ? cos( x)

Explanation:

Step1: Find the amplitude

The amplitude \(A\) of a cosine function \(y = A\cos(Bx)\) is the maximum value of \(y\). From the graph, the maximum value is \(1\) and the minimum is \(- 1\). The amplitude \(A=\frac{\text{Max}-\text{Min}}{2}=\frac{1 - (-1)}{2}=1\)

Step2: Find the period and then \(B\)

The general formula for the period \(T\) of \(y = A\cos(Bx)\) is \(T=\frac{2\pi}{B}\).
From the graph, the period \(T=\frac{2\pi}{3}\) (from \(x = 0\) to \(x=\frac{2\pi}{3}\) is one - full cycle).
Using the formula \(T=\frac{2\pi}{B}\), we substitute \(T = \frac{2\pi}{3}\)
\(\frac{2\pi}{3}=\frac{2\pi}{B}\), solving for \(B\) gives \(B = 3\)

Answer:

\(y = 1\cos(3x)\)