Transformations of Trigonometric Functions

Stretch v. Compression  

Transformations f(x) = a sin(bx – c) + k Stretch or Compression Vertical Shift Horizontal Shift, Phase Shift Period Shift Pay attention to the parentheses!!!

Examples Stretch/Compression: Horizontal Shift: Vertical Shift:

Sketching Transformations Step 1: Graph the parent equation Step 2: Identify the correct order of operations for the function Step 3: Make a table that follows the order of operations for the function Step 4: Complete the table for the key points (0, , , , )

*Memorize: y = sin(t) t sin(t)

Example 1: y = -2 sin(t) – 2 t

Example 2: t

Cosine Graph cos(t) = x t cos t

*Memorize: y = cos(t) t cos t

Transformations f(x) = a cos(bx – c) + k Stretch or Compression Vertical Shift Horizontal Shift, Phase Shift Period Shift Pay attention to the parentheses!!!

Cosine Examples Stretch/Compression: Horizontal Shift: Vertical Shift:

Example 1: y = –cos(t) – 1 t

Example 2: t

Transformations f(x) = a tan(bx – c) + k Stretch or Compression Vertical Shift Horizontal Shift, Phase Shift Period Shift Pay attention to the parentheses!!!

Tangent Graph tan(t) = x t tan t

Tangent Examples Stretch/Compression: Horizontal Shift:     Stretch/Compression: Horizontal Shift: Vertical Shift: Stretch/Compression: Horizontal Shift: Vertical Shift:

  x   tan x

  x   tan x

Transformations f(x) = a cot(bx – c) + k Stretch or Compression Vertical Shift Horizontal Shift, Phase Shift Period Shift Pay attention to the parentheses!!!

Cotangent Graph cot(t) = x t cot t

Cotangent Examples Stretch/Compression: Horizontal Shift:     Stretch/Compression: Horizontal Shift: Vertical Shift: Stretch/Compression: Horizontal Shift: Vertical Shift:

  x   tan x

  x   tan x

Transformations f(x) = a sec(bx – c) + k Stretch or Compression Vertical Shift Horizontal Shift, Phase Shift Period Shift Pay attention to the parentheses!!!

Examples Stretch/Compression: Horizontal Shift: Vertical Shift:

*Memorize: y = sec(t) t sec (t)

Example 1: y = -3 sec(t) – 1 t

Example 2: t

Transformations f(x) = a csc(bx – c) + k Stretch or Compression Vertical Shift Horizontal Shift, Phase Shift Period Shift Pay attention to the parentheses!!!

Examples Stretch/Compression: Horizontal Shift: Vertical Shift:

*Memorize: y = csc(t) t csc(t)

Example 1: y = -3 csc(πt) – 1 t

Example 2: t