y=5sin(8x+п/2)
The function you've provided is \( y = 5 \sin(8x + \frac{\pi}{2}) \).
This function is a sinusoidal function, and we can analyze its components:
Amplitude: The amplitude of the sine function is given by the coefficient in front, which is 5. This means the function will oscillate between -5 and 5.
Frequency: The frequency is determined by the coefficient of \( x \) inside the sine function. Here, the coefficient is 8, which means the frequency is \( 8 \) cycles per \( 2\pi \). The period \( T \) of the function can be calculated using the formula: \[ T = \frac{2\pi}{\text{frequency}} = \frac{2\pi}{8} = \frac{\pi}{4} \]
Phase Shift: The phase shift can be found from the expression inside the sine function. The phase shift is given by \( -\frac{C}{B} \) where the function is in the form \( y = A \sin(Bx + C) \). Here:
- \( A = 5 \)
- \( B = 8 \)
- \( C = \frac{\pi}{2} \)
Thus, the phase shift is: \[ -\frac{\frac{\pi}{2}}{8} = -\frac{\pi}{16} \] This indicates a shift to the left by \( \frac{\pi}{16} \).
Vertical Shift: There is no vertical shift in this function since there is no constant added or subtracted outside of the sine function.
Summary
- Amplitude: \( 5 \)
- Period: \( \frac{\pi}{4} \)
- Phase Shift: \( -\frac{\pi}{16} \) (to the left)
- Vertical Shift: \( 0 \)
You can sketch the graph based on these properties, marking the amplitude, period, and the phase shift.
