Let's find the equation of the sinusoidal function [tex]\( f(x) \)[/tex].
### Step-by-Step Solution
1. Determine the Midline (D):
The midline of a sinusoidal function is the horizontal line that represents the average value of the function. From the given information, the function intersects its midline at [tex]\(y = -2\)[/tex]. Therefore,
[tex]\[
D = -2
\][/tex]
2. Find the Amplitude (A):
The amplitude is the distance from the midline to the maximum or minimum point of the function. The minimum point is given as [tex]\(\left(\frac{3\pi}{2}, -7\right)\)[/tex]. The vertical distance from the midline [tex]\(y = -2\)[/tex] to the minimum point [tex]\(y = -7\)[/tex] is:
[tex]\[
A = |-7 - (-2)| = |-7 + 2| = | - 5| = 5
\][/tex]
3. Calculate the Period and Find B:
The period [tex]\(T\)[/tex] of a sinusoidal function is the distance required for the function to complete one full cycle. Since the minimum point occurs at [tex]\(x = \frac{3\pi}{2}\)[/tex], which represents half of the period (from midline to minimum is one-quarter of the period and from minimum back to midline is another quarter), the entire period is:
[tex]\[
T = 2 \times \frac{3\pi}{2} = 3\pi
\][/tex]
The period is related to the coefficient [tex]\(B\)[/tex] by the formula:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
Solving for [tex]\(B\)[/tex],
[tex]\[
B = \frac{2\pi}{T} = \frac{2\pi}{3\pi} = \frac{2}{3}
\][/tex]
4. Determine the Phase Shift (C):
The phase shift [tex]\(C\)[/tex] determines the horizontal translation of the function. Since the sinusoidal function intersects the midline at [tex]\(x = 0\)[/tex] and follows the form [tex]\( f(x) = A \sin(Bx + C) + D \)[/tex], we notice the sine function must start at zero. Hence,
[tex]\[
C = 0
\][/tex]
5. Form the Sinusoidal Function:
Now that we have [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex], we can write the equation of the function:
[tex]\[
f(x) = 5 \sin\left(\frac{2}{3}x\right) - 2
\][/tex]
Therefore, the formula of the sinusoidal function is:
[tex]\[
f(x) = 5 \sin\left(\frac{2}{3}x\right) - 2
\][/tex]
