question 6\nthe curve above is the graph of a sinusoidal function. it goes through the points (-8,0) and…

question 6\nthe curve above is the graph of a sinusoidal function. it goes through the points (-8,0) and (2,0). 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.\nf(x) = \nquestion help: message instructor post to forum\nsubmit question

question 6\nthe curve above is the graph of a sinusoidal function. it goes through the points (-8,0) and (2,0). 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.\nf(x) = \nquestion help: message instructor post to forum\nsubmit question

Answer

Explanation:

Step1: Find the amplitude (A)

The general form of a sinusoidal function is (y = A\sin(B(x - C))+D) or (y=A\cos(B(x - C)) + D). The amplitude (A) is the distance from the mid - line to the maximum (or minimum) value. The maximum value is (y = 2) and the minimum value is (y=-2), so (A = 2).

Step2: Find the mid - line (D)

The mid - line (D=\frac{2+( - 2)}{2}=0).

Step3: Find the period (T)

The period (T) is the distance between two consecutive (x) - values where the function has the same phase (e.g., two consecutive zeros). Given two zeros at (x=-8) and (x = 2), the period (T=2-( - 8)=10). Using the formula (T=\frac{2\pi}{B}), we solve for (B): (B=\frac{2\pi}{T}=\frac{2\pi}{10}=\frac{\pi}{5}).

Step4: Find the phase shift (C)

We can use the cosine function (y = A\cos(B(x - C))+D). Let's assume (D = 0), (A = 2), (B=\frac{\pi}{5}). When (x=-1), (y = 2) (a maximum). Substituting into (y = 2\cos(\frac{\pi}{5}(x - C))), we get (2=2\cos(\frac{\pi}{5}(-1 - C))), which simplifies to (\cos(\frac{\pi}{5}(-1 - C)) = 1). Then (\frac{\pi}{5}(-1 - C)=2k\pi), (k\in\mathbb{Z}). Taking (k = 0), we have (-1 - C=0), so (C=-1).

Answer:

(f(x)=2\cos\left(\frac{\pi}{5}(x + 1)\right))