Writing an Equation for a Graph of a Cosine Function

Identifying Key Components

Units

• Horizontal Units: Each unit is 1π
• Vertical Units: Each unit is 1

Amplitude

• The midline of the function is at y = -2.
• From the midline:
  • Minimum is 3 units down
  • Maximum is 3 units up
• Amplitude: 3
  • Leading coefficient a could be ±3 depending on the situation.
• Alternative method:
  • Measure entire vertical distance from max to min: 6 units
  • Divide by 2 to get amplitude: 3

Period Calculation

• Measure from peak to peak:
  • From one minimum to the next = 4 horizontal units = 4π
• Period: 4π
  • Period formula: 2π/B = 4π
  • Solving for B: B = 1/2

Determining Vertical Shift

• Original midline is the x-axis.
• Shift down by 2 units.
• Vertical Shift (K): -2

Analyzing Phase Shift

• At x = 0, the graph takes a maximum value.
• Cosine at x = 0 is also a maximum.
• Phase Shift (H): 0

Constructing the Equation

• Amplitude (A): Positive 3 (no reflection since the function starts at a maximum)
• B Value: 1/2
• Vertical Shift (K): -2
• No phase shift required.

Final Equation

• f(x) = 3 * cos(x/2) - 2
  • This incorporates all derived values (A, B, and K) with no phase shift necessary.*