Lecture: Sinusoidal Functions and Graph Transformations

Connection to Previous Course (Math 111)

• Importance of Math 111: Understanding from Math 111 aids smooth progression in Math 112.
• Review Recommended: Brush up on Math 111, especially graph transformations.
• Link Provided: Use the provided link to revise Math 111 notes for better understanding.

Key Concepts

• Graph Transformations: Understanding transformations from Math 111 applied to sine and cosine functions.
• Filling in the Table: Goal is to complete the provided table with transformation details.
• Focus on Sinusoidal Functions: Sinusoidal functions are sine functions that are transformed. Cosine functions are also sinusoidal due to their relationship with sine through shifting.

Sinusoidal Functions

• Definition: Transformed sine waves. Cosine functions are also sinusoidal because they can be derived by shifting sine functions.
• Graph Characteristics: Stretching, shifting, vertical distortions, etc.

Transformations Overview

• Vertical Stretching: Changing the amplitude by multiplying the output.
  • Example: y = 3 * sin(t) stretches vertically by 3, achieving outputs of ±3 instead of ±1.
• Vertical Shifting: Adding constants outside the function; affects midline.
  • Example: y = cos(t) + 2 shifts the graph up by 2 units.
• Horizontal Stretching/Compressing: Multiplying the input affecting period.
  • Example: y = cos(2t) compresses horizontally by a factor of ½, period becomes π.
• Horizontal Shifting: Adding/subtracting constants inside the function; affects phase start.
  • Example: y = cos(t - π/3) shifts the graph right by π/3 units.*

Detailed Transformations

Vertical Stretching

• Multiplication Outside the Function: y = a * sin(t) or y = a * cos(t).
  • Result: Amplitude becomes the constant 'a'.
  • Negative Multipliers: Reflects the graph vertically.
  • Activity Example: Adjust graphs and observe the effect of different 'a' values in Desmos.

Vertical Shifting

• Addition Outside the Function: y = sin(t) + k or y = cos(t) + k.
  • Result: Midline shifted to y = k.
  • Activity Example: Adjust vertical shifts in Desmos.

Horizontal Stretching/Compressing

• Multiplication Inside the Function: y = sin(Wt) or y = cos(Wt).
  • Result: Period becomes 2π/W. Horizontal compressing for W > 1 and stretching for W < 1.
  • Activity Example: Adjust W values in Desmos to see horizontal transformations.

Horizontal Shifting

• Addition/Subtraction Inside the Function: y = sin(t ± h) or y = cos(t ± h).
  • Result: Shifts the graph to the left (if +) or right (if -).
  • Activity Example: Adjust h values in Desmos to observe horizontal shifts.

Combined Transformations

• Multiplication and Addition Inside the Function: y = sin(W(t - h)) or y = cos(W(t - h)).
  • Remember to factor out W to correctly interpret h.
  • Example: y = sin(0.5t - π/4) becomes y = sin(0.5(t - π/2)) specifying the shift.
  • Comprehensive Activity: Adjust complex transformations in Desmos to see combined effects.

Drawing Sinusoidal Graphs

• Key Features to Determine: Amplitude, Midline, Period, and Phase (shift).
• Example Graph: y = -2sin(0.5t - π/4) + 3.
  • Amplitude: 2
  • Midline: y = 3
  • Period: 4π (calculated from 2π/0.5)
  • Phase Shift: Left by π/2 (from factoring 0.5)
• Drawing Tips: Start from the phase shift point, ensuring midline, amplitude, and period are correct.

Conclusion

• Review Key Facts: Use the table of transformations for quick referencing.
• Practical Applications: Practice drawing graphs to solidify understanding.

Resources

• Desmos Interactive Graphs: Links provided to visualize transformations dynamically.
• Math 111 Review: Brush up using the provided link for better comprehension of Math 112 material.