find an equation for the sine graph, f(x):\nwrite your answer in the form f(x) = a sin (bx + c) + d, where…

Explanation:

Step1: Find the amplitude A

The amplitude is half the vertical distance between the maximum and minimum values. The maximum value is - 2 and the minimum value is - 4. So, $A=\frac{|-2-(-4)|}{2}=\frac{2}{2} = 1$.

Step2: Find the vertical shift D

The vertical - shift D is the mid - value between the maximum and minimum values. $D=\frac{-2+( - 4)}{2}=\frac{-6}{2}=-3$.

Step3: Find the period and B

The period of a sine function is given by $T=\frac{2\pi}{|B|}$. From the graph, the period $T = \pi$. So, $\pi=\frac{2\pi}{|B|}$, which gives $|B| = 2$. Since the function is not horizontally reflected, $B = 2$.

Step4: Find the phase shift C

The standard sine function $y=\sin(x)$ has a zero - crossing at $x = 0$. For the given function, the zero - crossing closest to the origin and moving to the right is at $x=\frac{\pi}{4}$. For the function $y = \sin(Bx + C)$, when $y = 0$, $Bx+C = 0$. Substituting $x=\frac{\pi}{4}$ and $B = 2$, we get $2\times\frac{\pi}{4}+C = 0$, so $C=-\frac{\pi}{2}$.

Answer:

$f(x)=\sin(2x-\frac{\pi}{2})-3$