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A function, f(x) = Asin(Bx) + H, has the following properties:
• a period of 6,
• a minimum value of 2,
f(2.5) = 5, and
A, B, and H are all positive constants.
A:
B:
H:

Sagot :

xero099

The values of the parameters of the sinusoidal function are [tex]A = 2[/tex], [tex]B = \frac{\pi}{3}[/tex] and [tex]H = 4[/tex], respectively.

Procedure - Determination of the parameters for a sinusoidal function

Sinusoidal function model (Period, minimum value, given value)

According to this question, a sinusoidal function is described as following:

[tex]f(x) = \Delta y \cdot \sin \frac{2\pi\cdot x}{T} + y_{o}[/tex] (1)

Where:

  • [tex]x[/tex] - Independent variable
  • [tex]T[/tex] - Period
  • [tex]\Delta y[/tex] - Amplitude
  • [tex]y_{o}[/tex] - Midpoint

Determination of the period, amplitude and midpoint

By direct comparison, we have the following equivalencies:

[tex]A = \Delta y[/tex] (2)

[tex]B = \frac{2\pi}{T}[/tex] (3)

[tex]H = y_{o}[/tex] (4)

By (3) and [tex]T = 6[/tex], we have that:

[tex]B = \frac{2\pi}{6}[/tex]

[tex]B = \frac{\pi}{3}[/tex]

Then we find the remaining variables by using the following system:

[tex]2 = A \cdot \sin \left(\frac{3\pi}{2} \right) + H[/tex]

[tex]2 = -A + H[/tex] (5)

[tex]5 = A\cdot \sin \left[\left(\frac{\pi}{3} \right)\cdot (2.5)\right] + H[/tex]

[tex]5 = A\cdot \sin \frac{5\pi}{6} + H[/tex]

[tex]5 = \frac{1}{2}\cdot A + H[/tex] (6)

The solution of this system of equations is: [tex]A = 2[/tex], [tex]H = 4[/tex]

The values of the parameters of the sinusoidal function are [tex]A = 2[/tex], [tex]B = \frac{\pi}{3}[/tex] and [tex]H = 4[/tex], respectively.

To learn more on sinusoidal functions, we kindly invite to check this verified question: https://brainly.com/question/12078395

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