Lecture Notes: Graphing Trigonometric Functions

Introduction

• Focus on graphing trigonometric functions: sine and cosine.
• Understanding the shape and transformations of these functions.

Sine Function (sin x)

• Basic Shape: Sinusoidal wave (sine wave), one period ends at 2π.
• Negative Sine: Flips over the x-axis.
• Extending Beyond One Period: Graph continues infinitely in both directions.
• Graphing One Period: Break into four key points: 0, π/2, π, 3π/2.

Cosine Function (cos x)

• Basic Shape: Starts at the top, unlike sine which starts at the center.
• Negative Cosine: Starts at the bottom.
• Graphing One Period: Similar division into five key points for two periods.

Key Concepts

Amplitude

• Formula: y = A sin(Bx + C) + D
• Amplitude (A): Absolute value, indicates vertical stretch/compression.
• Example: For y = 2 sin x, amplitude = 2 (range from -2 to 2).

Period

• Formula: Period = 2π/B
• Impact of B: Changes horizontal stretch/compression of the wave.
• Example: y = sin 2x results in a period of π.

Graphing Examples

Sin and Cosine with Different Parameters

• Amplitude Changes: For instance, y = -3 cos x has amplitude 3.
• Graphing Steps:
  1. Determine amplitude and period.
  2. Divide each period into four intervals.
  3. Plot points and connect them.

Vertical Shift

• Graphs like sin x + 3 are vertically shifted by 3 units.
• Procedure:
  1. Determine new midline.
  2. Plot points around this new line.

Phase Shift

• Formula: Set Bx + C = 0 to find shift.
• Impact: Graph shifts left or right.
• Example: For y = sin(x - π/2), phase shift = π/2.

Additional Graphing Examples

Detailed Examples

• Function: 4 cos(πx)

  • Amplitude = 4, period = 2 (2π/π).
  • Plot from 0 to 2 and divide into intervals.

• Function: 2 sin(x - π/4) + 3

  • Amplitude = 2, vertical shift = +3.
  • Phase shift is π/4.
  • Plot vertically around y=3.

Domain and Range

• Domain: All real numbers for sine and cosine functions.
• Range: Determined by amplitude and vertical shifts.
  • Example: For y = -3 sin(x) + 4, range from 1 to 7.

Conclusion

• Understanding transformations helps in sketching accurate graphs.
• Practice with different values for amplitude, period, phase shifts, and vertical shifts.