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Sagot :
To determine the formula of the sinusoidal function given the minimum point and the midline intersection, we need to find the amplitude, period, phase shift, and vertical shift of the sine function. Here are the steps:
1. Determine the Vertical Shift:
The midline of the sinusoidal function is given by [tex]\( y = 5 \)[/tex]. Hence, the vertical shift [tex]\( D \)[/tex] is:
[tex]\[ D = 5 \][/tex]
2. Find the Amplitude:
The minimum point is at [tex]\( (0, 3) \)[/tex] and the midline is at [tex]\( y = 5 \)[/tex]. The amplitude [tex]\( A \)[/tex] is the distance from the midline to the minimum point. Thus:
[tex]\[ A = 5 - 3 = 2 \][/tex]
3. Calculate the Frequency and Period:
The function intersects its midline at [tex]\( (5\pi, 5) \)[/tex]. Given that [tex]\( (5\pi, 5) \)[/tex] represents a quarter of the period (since it is a transition from minimum to midline), the quarter period is:
[tex]\[ quarter\_period = 5\pi \][/tex]
Hence, the total period [tex]\( T \)[/tex] is four times the quarter period:
[tex]\[ T = 4 \times 5\pi = 20\pi \][/tex]
The frequency [tex]\( B \)[/tex] is calculated as [tex]\( \frac{2\pi}{T} \)[/tex]:
[tex]\[ B = \frac{2\pi}{20\pi} = 0.1 \][/tex]
4. Determine the Phase Shift:
The function is a sinusoidal function that reaches its minimum value at [tex]\( x = 0 \)[/tex]. For the standard sine function [tex]\( y = \sin(x) \)[/tex], the minimum occurs at [tex]\( x = -\frac{\pi}{2} \)[/tex]. Therefore, we need a phase shift [tex]\( C \)[/tex] to move this minimum to [tex]\( x = 0 \)[/tex].
So, we solve for [tex]\( C \)[/tex] in the equation:
[tex]\[ Bx + C = -\frac{\pi}{2} \quad \text{at} \quad x = 0 \][/tex]
[tex]\[ C = -\frac{\pi}{2} \][/tex]
5. Construct the Function:
Combining all these components, the formula for the sinusoidal function is:
[tex]\[ f(x) = A \sin(Bx + C) + D \][/tex]
Substituting the values [tex]\( A = 2 \)[/tex], [tex]\( B = 0.1 \)[/tex], [tex]\( C = -\frac{\pi}{2} \)[/tex], and [tex]\( D = 5 \)[/tex], the formula is:
[tex]\[ f(x) = 2 \sin(0.1x - \frac{\pi}{2}) + 5 \][/tex]
Thus, the function is:
[tex]\[ f(x) = 2 \sin(0.1 x - \frac{\pi}{2}) + 5 \][/tex]
1. Determine the Vertical Shift:
The midline of the sinusoidal function is given by [tex]\( y = 5 \)[/tex]. Hence, the vertical shift [tex]\( D \)[/tex] is:
[tex]\[ D = 5 \][/tex]
2. Find the Amplitude:
The minimum point is at [tex]\( (0, 3) \)[/tex] and the midline is at [tex]\( y = 5 \)[/tex]. The amplitude [tex]\( A \)[/tex] is the distance from the midline to the minimum point. Thus:
[tex]\[ A = 5 - 3 = 2 \][/tex]
3. Calculate the Frequency and Period:
The function intersects its midline at [tex]\( (5\pi, 5) \)[/tex]. Given that [tex]\( (5\pi, 5) \)[/tex] represents a quarter of the period (since it is a transition from minimum to midline), the quarter period is:
[tex]\[ quarter\_period = 5\pi \][/tex]
Hence, the total period [tex]\( T \)[/tex] is four times the quarter period:
[tex]\[ T = 4 \times 5\pi = 20\pi \][/tex]
The frequency [tex]\( B \)[/tex] is calculated as [tex]\( \frac{2\pi}{T} \)[/tex]:
[tex]\[ B = \frac{2\pi}{20\pi} = 0.1 \][/tex]
4. Determine the Phase Shift:
The function is a sinusoidal function that reaches its minimum value at [tex]\( x = 0 \)[/tex]. For the standard sine function [tex]\( y = \sin(x) \)[/tex], the minimum occurs at [tex]\( x = -\frac{\pi}{2} \)[/tex]. Therefore, we need a phase shift [tex]\( C \)[/tex] to move this minimum to [tex]\( x = 0 \)[/tex].
So, we solve for [tex]\( C \)[/tex] in the equation:
[tex]\[ Bx + C = -\frac{\pi}{2} \quad \text{at} \quad x = 0 \][/tex]
[tex]\[ C = -\frac{\pi}{2} \][/tex]
5. Construct the Function:
Combining all these components, the formula for the sinusoidal function is:
[tex]\[ f(x) = A \sin(Bx + C) + D \][/tex]
Substituting the values [tex]\( A = 2 \)[/tex], [tex]\( B = 0.1 \)[/tex], [tex]\( C = -\frac{\pi}{2} \)[/tex], and [tex]\( D = 5 \)[/tex], the formula is:
[tex]\[ f(x) = 2 \sin(0.1x - \frac{\pi}{2}) + 5 \][/tex]
Thus, the function is:
[tex]\[ f(x) = 2 \sin(0.1 x - \frac{\pi}{2}) + 5 \][/tex]
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