the graph of a sine function is drawn. one full cycle goes from x = 0 to x = 2π/5 and the low point on that…

the graph of a sine function is drawn. one full cycle goes from x = 0 to x = 2π/5 and the low point on that cycle is (π/10, - 3). which of the following functions could have this graph? y = 3 sin(1/5 x) y = 5 sin 3x y = 3 sin 5x y = - 3 sin 5x

the graph of a sine function is drawn. one full cycle goes from x = 0 to x = 2π/5 and the low point on that cycle is (π/10, - 3). which of the following functions could have this graph? y = 3 sin(1/5 x) y = 5 sin 3x y = 3 sin 5x y = - 3 sin 5x

Answer

Explanation:

Step1: Recall period formula

The period of the sine - function $y = A\sin(Bx)$ is $T=\frac{2\pi}{B}$. Given that the period $T = \frac{2\pi}{5}$. Set $\frac{2\pi}{B}=\frac{2\pi}{5}$, then $B = 5$.

Step2: Analyze amplitude and sign

The general form of a sine - function is $y=A\sin(Bx)$. The amplitude is $|A|$. The low - point of the sine function $y = A\sin(Bx)$ is $y=-|A|$. Given the low - point is $y = - 3$, so $|A| = 3$. Also, when $x=\frac{\pi}{10}$, $y=-3$. For $y = A\sin(Bx)$ with $B = 5$, when $x=\frac{\pi}{10}$, $y=A\sin(5\times\frac{\pi}{10})=A\sin(\frac{\pi}{2})$. Since $\sin(\frac{\pi}{2}) = 1$ and $y=-3$ at $x = \frac{\pi}{10}$, then $A=-3$.

Answer:

$y=-3\sin(5x)$