Graph of the Sine Function

1. Basic Shape

• The graph of the sine function, y = sin(x), is a wave-like curve that repeats itself indefinitely.

• It oscillates between -1 and 1.

2. Key Features

• Period: The length of one complete cycle of the wave. For the sine function, the period is 2π.

• Amplitude: The distance from the center line of the wave to its peak or trough. For the sine function, the amplitude is 1.

• Zeros: The points where the graph intersects the x-axis. For the sine function, the zeros occur at x = 0, π, 2π, 3π, and so on.

• Maximum and Minimum Points:

  • Maximum points: (π/2, 1), (5π/2, 1), (9π/2, 1), etc.

  • Minimum points: (3π/2, -1), (7π/2, -1), (11π/2, -1), etc.

3. Transformations

• Amplitude Changes:

  • y = A*sin(x):

    • If A > 1, the amplitude increases.

    • If 0 < A < 1, the amplitude decreases.

• Period Changes:

  • y = sin(Bx):

    • The period becomes 2π/B.

• Phase Shifts:

  • y = sin(x - C):

    • Shifts the graph horizontally by C units to the right.

• Vertical Shifts:

  • y = sin(x) + D:

    • Shifts the graph vertically by D units.

4. Applications

• The sine function is used to model many real-world phenomena that exhibit periodic behavior, such as:

  • Sound waves

  • Light waves

  • Alternating current (AC) electricity

  • Tides