f(x) is a sinusoidal curve with maximum at f(0)=10 and next minimum at f(6)=0. Find the equation?
To find the equation of the sinusoidal curve, we need to determine the amplitude, period, phase shift, and vertical shift.
First, let's determine the amplitude. The amplitude represents half the difference between the maximum and minimum values of the function. In this case, the maximum is f(0) = 10, and the next minimum is f(6) = 0. Therefore, the amplitude is (10 - 0) / 2 = 5.
Next, let's find the period. The period refers to the length of one complete cycle of the sinusoidal curve. To find it, we need to identify the distance between two consecutive maximum or minimum points. In this case, the maximum is at f(0) = 10, and the next minimum is at f(6) = 0. So, the period is 6 - 0 = 6.
Now, let's find the phase shift. The phase shift tells us the horizontal shift of the sinusoidal curve. Since the maximum occurs at f(0) = 10, the curve is not shifted horizontally. Therefore, the phase shift is 0.
Finally, let's find the vertical shift. The vertical shift represents the vertical displacement of the sinusoidal curve. In this case, the maximum is at f(0) = 10, which is above the x-axis. This indicates a positive vertical shift. Since the maximum is 10, the vertical shift is 10.
Now that we have determined the values for amplitude, period, phase shift, and vertical shift, we can write the equation of the sinusoidal curve. The general form for a sinusoidal curve is:
f(x) = A * sin(B(x - C)) + D
where:
A is the amplitude
B is the reciprocal of the period (B = 2π / period)
C is the phase shift
D is the vertical shift
Plugging in the values we found:
A = 5
B = 2π / 6
C = 0
D = 10
The equation of the sinusoidal curve is:
f(x) = 5 * sin((2π / 6)(x - 0)) + 10
Simplifying further,
f(x) = 5 * sin((π / 3)x) + 10