Question

How do you sketch a cycle of a sine curve with an amplitude of 1 and a period of 4pi, and how would you write the equation?

Answer 1

To sketch a cycle of a sine curve with an amplitude of 1 and a period of 4π, you can follow these steps:

1. Determine the key points: Identify the critical points on the curve, which include the maximum, minimum, and x-intercepts. For a sine curve, the maximum and minimum points occur at the amplitude value, and the x-intercepts occur at intervals of half the period.

   In this case, the maximum and minimum points are at y = ±1, and the x-intercepts are at x = 0, x = 2π, and x = 4π.

2. Plot the key points: On a graph paper or a coordinate plane, mark the key points you determined in the previous step.

   • The maximum point is at (0, 1).
   • The minimum point is at (2π, -1).
   • The next maximum point is at (4π, 1).
   • The x-intercepts are at (0, 0), (2π, 0), and (4π, 0).

3. Draw the curve: Connect the key points with a smooth curve that oscillates between the maximum and minimum points.

The equation for the sine curve with an amplitude of 1 and a period of 4π can be written as:

y = sin(x/2),

where x is the variable representing the angle in radians. This equation represents the standard form of a sine curve with adjustments for amplitude and period.