Graphing Trigonometric Functions

Sine Function

• The sine function ( \sin(x) ) is a sinusoidal function that forms a sine wave.
• One Period: Ends at ( 2\pi ), representing one cycle.
• Negative Sine: Flips over the x-axis; starts downward from origin.
• Infinite Extension: The wave extends indefinitely in both directions.
• Key Points in One Cycle:
  • Break into four points: ( \pi/2, \pi, 3\pi/2 ).

Cosine Function

• The cosine function ( \cos(x) ) starts at the top of the wave, unlike sine which starts at the center.
• Negative Cosine: Starts at the bottom, goes up, and then back down.
• Graphing Two Cycles:
  • Divide each cycle into five key points.
  • Period: ( 2\pi ).

Amplitude and Period

• Generic Formula: ( a\sin(bx + c) + d )
• Amplitude (a): Distance from midline to peak, affects vertical stretch/compression.
  • Example: ( 2\sin(x) ) varies from 2 to -2.
• Period Calculation: ( \text{Period} = \frac{2\pi}{b} )

Effects of Amplitude and Period

• Graph Variations:
  • ( 2\sin(1/2x) ): Amplitude is 2, period is ( 4\pi ).
  • ( 4\cos(\pi x) ): Amplitude is 4, period is 2.
• Domain and Range:
  • Domain of sine/cosine: all real numbers.
  • Range determined by amplitude (e.g., from -4 to 4).

Vertical Shifts

• Example: ( \sin(x) + 3 )
  • Vertical shift moves the wave up to new midline at y=3.
• Graphing Steps:
  • Plot vertical shift as horizontal line.
  • Determine new range based on amplitude.

Phase Shifts

• Identifying Phase Shift: Set the inside of the sine function equal to zero.
  • Example shift for ( \sin(x - \pi/2) ) is ( \pi/2 ).
• Graphing With Phase Shift:
  • Starts at phase shift point, not origin.
  • Example: ( 2\sin(x - \pi/4) + 3 )
  • Vertical shift: 3, amplitude: 2, phase shift: ( \pi/4 ).

Practice Examples

• Graph ( \sin(x - \pi/2) ):

  • Amplitude: 1, period: ( 2\pi ).
  • Phase starts at ( \pi/2 ).

• Graph ( 2\sin(x - \pi/4) + 3 ):

  • Vertical shift: 3, amplitude: 2.
  • Range: 1 to 5.
  • Phase shift from ( \pi/4 ), period: ( 2\pi ).

Conclusion

• Trigonometric functions use amplitude, period, and phase/vertical shifts to determine waveform characteristics.
• Accurate graphing requires understanding these parameters and key points within cycles.