a sinusoidal function has a local maximum at (0,2) and the next minimum at (Pi/4, -2). a correct equation for the function is
max at t = 0 is a cos function
y = A cos (k theta)
max is 2 so
y = 2 cos (k theta)
min of cos is at (k *pi/4) = pi radians
so k = 4
y = 2 cos (4 theta)
Oh if they insist on sin rather than cos, use phase shift of pi/2
y = 2 sin (4 theta - pi/2)
not -pi/2 but +pi/2
To find the equation of the sinusoidal function given the local maximum and minimum, we need to determine the amplitude, period, phase shift, and vertical shift.
1. Amplitude: The amplitude of the function is the distance between the centerline (the average value) and the peak or trough. In this case, the local maximum is at (0, 2), so the amplitude is 2.
2. Period: The period is the distance between two consecutive peaks or troughs. We know that the next minimum is at (π/4, -2), which indicates that one full cycle of the function occurs from x = 0 to x = π/4. Therefore, the period is π/4.
3. Phase Shift: The phase shift determines how the graph is shifted to the right or left. Since the local maximum occurs at x = 0, there is no horizontal shift. Therefore, the phase shift is 0.
4. Vertical Shift: The vertical shift determines how the graph is shifted up or down. The local maximum is at y = 2, so the vertical shift is 2.
Putting all this information together, the equation for the sinusoidal function is:
y = A * sin(B * (x - C)) + D
Substituting the values we determined:
y = 2 * sin(4 * (x - 0)) + 2
Simplifying further:
y = 2 * sin(4x) + 2
Thus, the correct equation for the given sinusoidal function is y = 2 * sin(4x) + 2.