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 distance from the mid - line to the peak (or trough). Since the maximum value is \(\frac{1}{3}\) and the minimum value is \(-\frac{1}{3}\), \(A=\frac{1}{3}\).

Step2: Determine the period \(T\)

The period \(T\) is the length of one full cycle. From the graph, \(T = 1\).

Step3: Calculate the angular frequency \(\omega\)

The formula for the period of a sine or cosine function is \(T=\frac{2\pi}{\omega}\). Given \(T = 1\), we solve for \(\omega\):

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Step4: Check if it is a sine or cosine function

The graph passes through the origin \((0,0)\). The general form of a sine function \(y = A\sin(\omega x)\) passes through \((0,0)\) (since \(y(0)=A\sin(0) = 0\)), while the general form of a cosine function \(y=A\cos(\omega x)\) has \(y(0)=A\). So, the function is a sine function.

Answer:

\(y=\frac{1}{3}\sin(2\pi x)\)