Graph of the Cosine Function

1. Basic Shape

• The graph of the cosine function, y = cos(x), is a wave-like curve that repeats indefinitely, similar to the sine function.

• It oscillates between -1 and 1.

2. Key Features

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

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

• Zeros: The points where the graph intersects the x-axis. For the cosine function, the zeros occur at x = π/2, 3π/2, 5π/2, and so on.

• Maximum and Minimum Points:

  • Maximum points: (0, 1), (2π, 1), (4π, 1), etc.

  • Minimum points: (π, -1), (3π, -1), (5π, -1), etc.

3. Transformations

• Amplitude Changes:

  • y = A*cos(x):

    • If A > 1, the amplitude increases.

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

• Period Changes:

  • y = cos(Bx):

    • The period becomes 2π/B.

• Phase Shifts:

  • y = cos(x - C):

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

• Vertical Shifts:

  • y = cos(x) + D:

    • Shifts the graph vertically by D units.

4. Relationship to Sine Function

• The cosine function is essentially a phase-shifted sine function.

  • cos(x) = sin(x + π/2)

• This means the cosine graph is the same as the sine graph shifted π/2 units to the left.

5. Applications

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

  • Sound waves

  • Light waves

  • Simple harmonic motion (e.g., the motion of a pendulum)