Graphing Sine and Cosine Functions with Shifts

Basic Structures

• Sine Function: $y = \sin x$

  • Amplitude: 1
  • Period: $2\pi$
  • Graph: Starts at center, goes up, back to middle, down, and returns to middle.
  • Range: Varies from [-1, 1]

• Negative Sine Function: $y = -\sin x$

  • Graph: Starts downward instead of upward.
  • Period: $2\pi$, Amplitude: 1

• Cosine Function: $y = \cos x$

  • Graph: Starts at top, then goes down, returns up.
  • Period: $2\pi$, Varies between [-1, 1].

• Negative Cosine Function: $y = -\cos x$

  • Graph: Starts at bottom and rises.

Modified Amplitude and Period

• Amplitude Change:

  • Example: $y = 2\sin x$
  • Amplitude becomes 2; range [-2, 2].

• Period Change with Frequency:

  • Example: $y = \sin 2x$
  • Horizontal shrink, Period: $\pi$ (calculated as $\frac{2\pi}{B}$).

General Equation

• Generic Equation for Sine/Cosine: $y = a\sin(Bx + C) + D$
  • $a$: Amplitude
  • $B$: Affects period ($\frac{2\pi}{B}$)
  • $C$: Phase shift (horizontal shift)
  • $D$: Vertical shift

Examples with Amplitude, Period, and Shifts

• Horizontal Stretch:

  • Example: $y = -\sin\frac{1}{2}x$
  • Horizontal stretch by factor of 2, Period: $4\pi$.

• Vertical and Horizontal Stretch:

  • Example: $y = 3\sin\frac{1}{3}x$
  • Amplitude: 3, Period: $9\pi$.

Vertical and Phase Shifts

• Vertical Shift:

  • Example: $y = \sin x + 2$
  • Center line at y=2, Amplitude: 1

• Combined Shifts & Example:

  • Example: $y = 2\sin(4x) - 3$
  • Vertical shift to $-3$, Amplitude: 2, Period: $\frac{\pi}{2}$.

• Phase Shift Calculation:

  • Set $Bx + C = 0$, solve for $x$ to find starting point.

Complex Graphs with All Shifts

• Example: $y = -3\cos\left(\frac{1}{2}x + \pi\right) + 5$
  • Vertical shift to 5, Amplitude: 3
  • Phase shift: -$2\pi$
  • Period: $4\pi$

Graphing Approach

1. Identify Vertical Shift: Plot midline.
2. Calculate and set Amplitude: Determine range.
3. Determine Phase Shift: Solve for start of cycle.
4. Calculate Period: Define complete cycle length.
5. Plot Points: Divide the period into four intervals.
6. Graph Shape: Follow sine/cosine starting point rules.

Important Details

• Domain: Generally $(-\infty, \infty)$ unless restricted.
• Range: From minimum to maximum values based on vertical shift and amplitude.

This guide summarizes graphing sine and cosine functions including amplitude, period, vertical, and phase shifts. Understanding these elements allows for accurate graph plotting of trigonometric functions.