write the equation of the trigonometric function shown in the graph.\nanswer attempt 1 out of 4

Explanation:

Step1: Identify function form

We use the general sinusoidal form: $y = A\sin(Bx + C) + D$

Step2: Find vertical shift (D)

The midline is the average of max (3) and min (-5): $D = \frac{3 + (-5)}{2} = \frac{-2}{2} = -1$

Step3: Find amplitude (A)

Amplitude is distance from midline to max: $A = 3 - (-1) = 4$

Step4: Find period and B

The period is the distance between two consecutive zeros (e.g., $0$ to $8\pi$), so period $T = 8\pi$. Using $B = \frac{2\pi}{T}$: $B = \frac{2\pi}{8\pi} = \frac{1}{4}$

Step5: Find phase shift (C)

The graph crosses the midline $y=-1$ at $x=0$, and increases upwards, matching a sine function with no phase shift. For $y = A\sin(Bx) + D$, substitute $x=0,y=-1$: $-1 = 4\sin(0) + D$, which confirms $D=-1$ and $C=0$.

Answer:

$y = 4\sin\left(\frac{1}{4}x\right) - 1$