Transformations of Trigonometric Functions

Overview

• Review of transformations from Grade 11
• Focus on transformations applied to sine and cosine functions
• Function form: ( K(X - D) + C )

Transformations

Y-coordinate Changes

• Vertical Stretch/Compression:
  • ( a ) affects amplitude
  • Stretch if ( a > 1 )
  • Compression if ( 0 < a < 1 )
  • Reflection if ( a < 0 )
• Vertical Shift:
  • ( C ) causes the entire function to shift up or down
  • Affects the axis placement

X-coordinate Changes

• Horizontal Stretch/Compression:
  • ( K ) affects period of the function
  • Period formula: ( \text{Period} = \frac{2\pi}{K} )
  • Compression if ( K > 1 )
  • Stretch if ( 0 < K < 1 )
• Horizontal Shift:
  • ( D ) shifts function left or right
  • Important to factor out ( K )

Example Problems

1. Graphing ( y = \frac{1}{2} \cos \left( \frac{x}{2} - \frac{\pi}{12} \right) - 1 )

   • Transformations:
     • ( Y ): ( \frac{1}{2}Y - 1 )
     • ( X ): ( 2X + \frac{\pi}{6} )
   • Period Calculation:
     • ( \text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi )
   • Key Points and Sketching:
     • Start by identifying key points and apply transformations.

2. Graphing ( y = -\sin(2(x + \frac{\pi}{4})) + 2 )

   • Transformations:
     • Axis: ( y = 2 )
     • Negative Sine Function: Reflects vertically
   • Period Calculation:
     • ( \text{Period} = \frac{2\pi}{2} = \pi )
   • Sketching Technique:
     • Identify axis and amplitude, then locate quarter points based on period

Key Concepts

• Understand how amplitude, period, and shifts affect sine and cosine graphs
• Factor out ( K ) for correct transformations
• Use both mapping rules and graphical techniques to plot functions accurately

Upcoming Lessons

• Word problems and applications involving these transformations
• Reciprocal trigonometric functions
• Modeling with trigonometric functions