Question

write an equation of the form $y = a \sin bx$ or $y = a \cos bx$ to describe the graph below.

Explanation:

Step 1: Determine the amplitude \(a\)

The amplitude \(a\) is the maximum distance from the mid - line to the peak (or trough) of the sine or cosine wave.
Looking at the graph, the maximum value \(y = 2\) and the minimum value \(y=-2\). The formula for the amplitude \(a=\frac{\text{Max}-\text{Min}}{2}\). So, \(a = 2\)

Step 2: Determine the period \(T\) and the value of \(b\)

The general formula for the period of a function \(y = a\sin(bx)\) or \(y=a\cos(bx)\) is \(T=\frac{2\pi}{b}\).
From the graph, the period \(T=\pi\).
Using the formula \(T = \frac{2\pi}{b}\), and substituting \(T=\pi\), we get \(\pi=\frac{2\pi}{b}\).
Solving for \(b\):

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Step 3: Determine the type of the function (\(\sin\) or \(\cos\))

The standard cosine function \(y = \cos(x)\) has a maximum at \(x = 0\). Looking at the given graph, when \(x = 0\), \(y=2\) (a maximum). So the function is of the form \(y=a\cos(bx)\)

Answer:

\(y = 2\cos(2x)\)