Respuesta :
The graph and therefore, the sinusoidal function equation is a wave function graph and equation
The formula of the sinusoidal function can be written as follows;
[tex]y = \mathbf{5 \cdot sin\left (\dfrac{1}{2} \cdot \left (x + \dfrac{\pi}{2} \right)\right)}[/tex]
The process of deriving the equation of the sinusoidal function is as follows:
The given parameters of the sinusoidal function are;
The coordinate of the maximum point = (0, 5)
The coordinate of the minimum point = (2·π, -5)
Method:
The general form of the sinusoidal function is y = A·sin(B·(x - h)) + k
From the given coordinates of points on the graph of the sinusoidal wave, the values of A, B, h, and k can be calculated as follows
Solution:
y = A·sin(B·(x - h)) + k
Where;
A = Amplitude = (Maximum y-value - Minimum y-value)/2
∴ A = (5 - (-5))/2 = 5
A = 5
The period, T = 2·π/B = The time to complete a cycle = Time from maximum y-value to another maximum y-value
∴ The period, T = 2 × (The x-value at maximum y-value - The x-value at minimum y-value)
T = 2 × (0 - 2·π) = -4·π
Given that the period, T represent a scalar quantity, we have T = 4·π
∴ T = 4·π = 2·π/B
B = 2·π/4·π = 1/2
B = 1/2
From the given coordinates, at x = 0, f(x) is maximum, therefore, sin(B·(x - h)) = 1, which gives;
B·(x - h) = π/2
(1/2)·(0 - h) = π/2
h = -π/2
The vertical shift, k = (Maximum y-value + Minimum y-value)/2
∴ k = (5 + (-5))/2 = 0, there is no vertical shift
Therefore;
The equation of the sinusoidal function is y = 5·sin(1/2·(x - (-π/2)) + 0, from which the equation can be presented as follows;
[tex]y = \mathbf{ 5 \cdot sin\left (\dfrac{1}{2} \cdot \left (x + \dfrac{\pi}{2} \right)\right)}[/tex]
Learn more about sinusoidal functions here
https://brainly.com/question/22706973
