Graphing Sine and Cosine Functions with Phase Shifts

Basic Structure

• Basic Equation:
  • y = sin(x)
• Amplitude: 1
• Period: 2π
  • Sine starts at the center, goes up, back to the middle, down, and then back to the middle.
  • Amplitude varies from 1 to -1.

Variations of Sine and Cosine

• Negative Sine (y = -sin(x))
  • Graph starts downwards instead of upwards.
• Cosine (y = cos(x))
  • Starts at the top, goes down, back up.
  • Varies between 1 and -1 with a period of 2π.
• Negative Cosine (y = -cos(x))
  • Graph starts at the bottom.

Graphing with Vertical and Horizontal Stretches/Shrinks

• Vertical Stretch Example (y = 2sin(x))
  • Regular sine wave, but amplitude is 2, varying from 2 to -2.
• Horizontal Shrink Example (y = sin(2x))
  • Amplitude is 1, period is π (2π/b where b = 2).

Generic Equation

• Equation: y = asin(bx + c) + d
  • a: Amplitude
  • b: Defines period
  • c: Horizontal/phase shift
  • d: Vertical shift

Examples and Calculations

Example 1: y = -sin(1/2x)

• Negative sine starts down.
• Horizontal stretch by a factor of 2.
• Period: 4π (2π / 0.5).

Example 2: y = 3cos(1/3x)

• Amplitude: 3
• Period: 9π (3π / (1/3)).

Introducing Vertical and Phase Shifts

Vertical Shift Example: y = sin(x) + 2

• Vertical shift: 2 (plot center line at 2).
• Amplitude: 1, range from 1 to 3.

Phase Shift Example: y = 2sin(4x - 3)

• Vertical shift: -3
• Amplitude: 2
• Phase shift calculation: Use 2π/b to find period.

Combined Vertical and Phase Shift Examples

Example 1: y = -3cos(1/2x + 5)

• Vertical shift: 5
• Amplitude: 3
• Period: 4π

Example 2: y = 2sin(x - π/2) + 3

• Vertical shift: 3
• Amplitude: 2
• Phase shift: π/2
• Period: 2π

Concepts

• Domain: generally from -∞ to ∞ unless restricted.
• Phase Shift Calculation: Set inside of the function = 0 and solve for x.

Conclusion

• Understanding how to graph with amplitude, vertical shift, and phase shift.
• Each element (a, b, c, d) influences the graph differently.

Practice

• Try graphing different combinations of sine and cosine functions with varying a, b, c, d to solidify understanding.