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Identify the amplitude, period, and phase shift, and graph the function. y=4sin(⁽ˣ ⁺ π/²⁾

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Final answer:

The function y=4sin(x + π/2) has an amplitude of 4 units, a period of 2π radians, and a phase shift of π/2 radians to the left. To graph the function, mark the amplitude and trough on the y-axis and draw the sine curve starting from the peak due to the phase shift.

Step-by-step explanation:

To identify the amplitude, period, and phase shift of the sinusoidal function y=4sin(x + π/2), we need to rewrite it to match the standard form y = A sin(Bx - C) + D. Here, A represents the amplitude, (2π)/B is the period, and C/B is the phase shift. Comparing the given function to the standard form:

  • The amplitude A is 4, which represents the maximum height of the wave from its mean position.
  • The period can be found by setting B equal to 1 (since it's not explicitly shown in the equation), which results in a period of 2π.
  • The phase shift is C/B, which translates to π/2 divided by 1, or simply π/2 radians. This indicates that the wave is shifted to the left by π/2 radians from the origin.

When graphing this function, start by plotting the midpoint of the wave on the y-axis (which is 0 for a sine function without vertical shift), then mark the peak amplitude at y = 4 and the trough at y = -4. Draw one complete cycle of the sine wave ensuring that the wave starts at its highest point because of the π/2 phase shift. Repeat the pattern to fill the graph.

User Sean Hagen
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