Question

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write the equation of a sine function with amplitude = 8 and period = 4π. type the equation in the form y = a sin (ωx) or y = a cos (ωx) with ω > 0. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. there is one equation. it is y = (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) b. there are two equations; the equation when a < 0 is y = and the equation when a > 0 is y = (simplify your answers. type exact answers, using π as needed. use integers or fractions for any numbers in the expressions.) c. there are no equations with these characteristics.

Explanation:

Step1: Recall the formula for the amplitude and period of a sine function

The general form of a sine function is \(y = A\sin(\omega x)\). The amplitude is \(|A|\) and the period is \(T=\frac{2\pi}{\omega}\).
Given that the amplitude \(|A| = 8\), so \(A=\pm8\). Given that the period \(T = 4\pi\).

Step2: Solve for \(\omega\)

Using the period formula \(T=\frac{2\pi}{\omega}\), substitute \(T = 4\pi\) into it:

$$
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Step3: Write the equations

When \(A = 8\), the equation is \(y = 8\sin(\frac{1}{2}x)\). When \(A=- 8\), the equation is \(y=-8\sin(\frac{1}{2}x)\)

Answer:

B. There are two equations; the equation when \(A < 0\) is \(y=-8\sin(\frac{1}{2}x)\) and the equation when \(A>0\) is \(y = 8\sin(\frac{1}{2}x)\)