Question

find an equation for the graph shown.
type the equation in the form ( y = a sin ( omega x ) + b ) or ( y = a cos ( omega x ) + b ).
( y = ) (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 displacement from the mid - line. The mid - line is \(y = 0\) (since the graph is symmetric about the origin). The maximum value of \(y\) (or minimum in magnitude) is \(A=\frac{8}{5}\).

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

The standard form of a sine function is \(y = A\sin(\omega x)+B\). For a sine function \(y=\sin(x)\), the period \(T = 2\pi\) and \(\omega=\frac{2\pi}{T}\).
From the graph, if we consider the basic sine - wave - like pattern (a sine function \(y = A\sin(\omega x)\) passes through the origin). The period \(T = 2\) (the distance between two consecutive points where the graph crosses the \(x\) - axis in the same increasing/decreasing trend). Using the formula \(\omega=\frac{2\pi}{T}\), with \(T = 2\), we get \(\omega=\pi\).

Step3: Write the equation of the sine function

Since \(B = 0\) (mid - line \(y = 0\)) and \(A=\frac{8}{5}\), \(\omega=\pi\), the equation of the function is \(y=\frac{8}{5}\sin(\pi x)\)

Answer:

\(y=\frac{8}{5}\sin(\pi x)\)