question 4 the curve above is the graph of a sinusoidal function. it goes through the points ( - 5, 0) and…

question 4 the curve above is the graph of a sinusoidal function. it goes through the points ( - 5, 0) and (1, 0). find a sinusoidal function that matches the given graph. if needed, you can enter π =3.1416... as pi in your answer, otherwise use at least 3 decimal digits. f(x) =

question 4 the curve above is the graph of a sinusoidal function. it goes through the points ( - 5, 0) and (1, 0). find a sinusoidal function that matches the given graph. if needed, you can enter π =3.1416... as pi in your answer, otherwise use at least 3 decimal digits. f(x) =

Answer

Explanation:

Step1: Find the period

The distance between two consecutive x - intercepts $(-5,0)$ and $(1,0)$ is half of the period. So, $T/2=1 - (-5)=6$, then $T = 12$. Using the formula $T=\frac{2\pi}{|B|}$, we have $12=\frac{2\pi}{|B|}$, so $|B|=\frac{\pi}{6}$. Let's assume $B=\frac{\pi}{6}$ for a standard - form sinusoidal function.

Step2: Determine the amplitude

The maximum value of the function is $y = 4$ and the minimum value is $y=-4$, so the amplitude $A = 4$.

Step3: Find the phase - shift and vertical - shift

The mid - line of the function is $y = 0$, so $D = 0$. The function passes through $(-5,0)$ and $(1,0)$. A sine function $y = A\sin(B(x - C))+D$. Let's use the point $(-5,0)$. Substituting $A = 4$, $B=\frac{\pi}{6}$, $x=-5$ and $y = 0$ into $y = A\sin(B(x - C))+D$: [0 = 4\sin\left(\frac{\pi}{6}(-5 - C)\right)] [\sin\left(\frac{\pi}{6}(-5 - C)\right)=0] (\frac{\pi}{6}(-5 - C)=k\pi), (k\in\mathbb{Z}). Let (k = 0), then (-5 - C=0), (C=-5). So the sinusoidal function is (y = 4\sin\left(\frac{\pi}{6}(x + 5)\right))

Answer:

(4\sin\left(\frac{\pi}{6}(x + 5)\right))