Understanding Sinusoidal Graphs: Sine vs. Cosine Equations

Introduction

• Sinusoidal graphs: S-shaped graphs commonly represented using sine or cosine equations.
• Common question: Should you use a sine or cosine equation?
  • Answer: Either can be used.
  • Positive or negative 'a' value? Either can be used.

Midline and Vertical Shift

• Midline: Splits the graph in half, indicating vertical shift.
• Example: If the graph shifts down by 2 units, the vertical shift (k value) is -2.

Amplitude

• Amplitude measures wave height from the midline.
• Always positive unless reflected over the x-axis.
• Example: Amplitude is 1.

Sine vs. Cosine Graphs

• Choice depends on the starting point.
• Basic Sine Graph:
  • Starts at the origin.
  • Pattern: Up, back to zero, down, back up.
• Basic Cosine Graph:
  • Starts at maximum, goes through zero, minimum, back to maximum.
• Both have a period of 2π.

Calculating the Period

• Measure from start to repeat point (e.g., 5π - 1π = 4π).
• Period = 4π.
• Formula: Period = 2π / B or B = 2π / Period.
• Example: B = 0.5.

Writing Equations

• Amplitude = 1, B = 0.5, Vertical Shift = -2.
• Determining sine or cosine depends on phase shift and reflection.

Cosine Equation Example

• Reflecting cosine graph over x-axis.
  • Equation: -1 cos(0.5x)
  • No horizontal shift (H = 0).

Sine Equation Example

• Sine graph shifts to the right by π.
  • Equation: sin(0.5(x - π))
  • No reflection, amplitude = 1.

Negative Sine Example

• Reflect sine graph over x-axis.
  • Shift to the left by π.
  • Equation: -sin(0.5(x + π))

Positive Cosine Example

• Using a point on the max, shifted left 2π.
  • Equation: cos(0.5(x + 2π))

Selecting the Appropriate Equation

• Teacher may specify sine or cosine.
• Choose minimal or no shift for simplicity.
• Example: Easier to use a cosine graph reflected and shifted down.

Conclusion

• Flexibility in choosing sine or cosine based on graph behavior.
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