5.3 Trigonometric Graphs 006

Section 5.3: Trigonometric Graphs

Overview

  • Exploration of:

    • Graphs of Sine and Cosine functions.

    • Transformations of Sine and Cosine functions.

Definition of Periodic Functions

  • Periodic Function: A function ( f , ) is periodic if there exists a positive number ( p ) such that:

    • ( f(t + p) = f(t) ) for every ( t ).

  • Period: The least positive number ( p ) if it exists.

Key Graph Characteristics

  • Sine Function:

    • Domain: All real numbers.

    • Range: ([-1, 1]).

    • Period: (2\pi).

    • Symmetry: Odd function, symmetric about the origin.

  • Cosine Function:

    • Domain: All real numbers.

    • Range: ([-1, 1]).

    • Period: (2\pi).

    • Symmetry: Even function, symmetric about the y-axis.

Graphs of Transformations of Sine and Cosine

General Form

  • For functions of the form: ( y = a \sin(kx) ) or ( y = a \cos(kx) )

    • Amplitude: The maximum value attained by the function; given by ( |a| ).

    • Period: Calculated as ( \frac{2\pi}{|k|} ).

Effects of Transformations

  1. Vertical Stretch/Shrink:

    • If ( a > 1 ), graph is vertically stretched.

    • If ( a < 1 ), graph is vertically shrunk.

  2. Reflection Over X-axis:

    • Occurs when ( a < 0 ).

  3. Horizontal Stretch/Shrink:

    • If ( k < 1 ), graph is horizontally stretched.

    • If ( k > 1 ), graph is horizontally shrunk.

  4. Key Points for Graphing:

    • Plot 5 key points: x-intercepts, maximum, minimum (peaks, valleys).

    • For sine: ( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi ) corresponding y-values are ( 0, 1, 0, -1, 0 ).

    • For cosine: ( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi ) corresponding y-values are ( 1, 0, -1, 0, 1 ).

Example 1: Sketching Transformations

  • Function: ( f(x) = -\sin(2x) )

    • Here, ( a < 0 ) indicates reflection about the x-axis; ( k = 2 ) means period is ( \frac{2\pi}{2} = \pi ).

    • Key points: Plot the transformed points based on calculations.

Example 2: Horizontal Shifts in Sine and Cosine

  • Functions: ( g(x) = \cos(6\pi x) )

    • Amplitude: ( |a| )

    • Period: ( \frac{2\pi}{6\pi} = \frac{1}{3} )

    • Horizontal shift: Occurs based on argument adjustment in the function.

Example 3: Vertical Shifts

  • Drawing the graph of ( f(x) = 2 + \cos(x) ):

    • Vertical shift by 2 units upward.

    • Domain: All real numbers.

    • Range: ([1, 3]) (Shift Range based on vertical transformation).

Conclusion

  • Understanding the transformations of trigonometric functions aids in sketching their graphs effectively, which incorporates shifts, stretches, and reflections.