Question

find an equation for the graph shown to the right.
type the equation in the form ( y = a sin (omega x) ) or ( y = a cos (omega x) ).
( y=square )
(type an exact answer, using ( pi ) as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Determine the amplitude \(A\)

The amplitude \(A\) is the maximum value of the function. From the graph, the maximum value is \(4\), so \(A = 4\).

Step2: Determine the period \(T\) and the angular frequency \(\omega\)

The period \(T\) is the length of one full cycle. From the graph, \(T=6\) (from \(x = - 3\) to \(x=3\)).
The formula for the period of \(y = A\cos(\omega x)\) is \(T=\frac{2\pi}{\omega}\).
Substituting \(T = 6\) into \(T=\frac{2\pi}{\omega}\), we get \(6=\frac{2\pi}{\omega}\).
Solving for \(\omega\), we multiply both sides by \(\omega\): \(6\omega=2\pi\), then \(\omega=\frac{\pi}{3}\).

Step3: Write the equation

Since the graph has a maximum at \(x = 0\), the function is a cosine function (because \(y=\cos(x)\) has a maximum at \(x = 0\)).
The equation is \(y = 4\cos(\frac{\pi}{3}x)\)

Answer:

\(y = 4\cos(\frac{\pi}{3}x)\)