Answer: [tex]\bold{A)\quad y=5sin\bigg(\dfrac{6}{5}x-\pi\bigg)-4}[/tex]
Step-by-step explanation:
The general form of a sin/cos function is: y = A sin/cos (Bx-C) + D where the period (P) = 2π ÷ B
In the given function, [tex]B=\dfrac{6}{5}[/tex] → [tex]P=2\pi \cdot \dfrac{5}{6}=\dfrac{10\pi}{3}[/tex]
Half of that period is: [tex]\dfrac{1}{2}\cdot \dfrac{10\pi}{3}=\large\boxed{\dfrac{5\pi}{3}}[/tex]
Calculate the period for each of the options to find a match:
[tex]A)\quad B=\dfrac{6}{5}:\quad 2\pi \div \dfrac{6}{5}=2\pi \cdot \dfrac{5}{6}=\dfrac{5\pi}{3}\quad \leftarrow\text{THIS WORKS!}\\\\\\B)\quad B=\dfrac{6}{10}:\quad 2\pi \div \dfrac{6}{10}=2\pi \cdot \dfrac{10}{6}=\dfrac{10\pi}{3}\\\\\\C)\quad B=\dfrac{5}{6}:\quad 2\pi \div \dfrac{5}{6}=2\pi \cdot \dfrac{6}{5}=\dfrac{12\pi}{5}\\\\\\D)\quad B=\dfrac{3}{10}:\quad 2\pi \div \dfrac{3}{10}=2\pi \cdot \dfrac{10}{3}=\dfrac{20\pi}{3}[/tex]
