Lesson 7.4: Graphing Sine and Cosine Functions

Essential Question

• How can you identify key features of sine and cosine functions?

Overview

• Focus on equations:
  • $y = a \sin(bx)$
  • $y = a \cos(bx)$
• Examine how coefficients a and b affect the graph

Key Features of Graphs

Amplitude

• Definition: The maximum distance a point on the graph is from the midline.
• Calculated as the absolute value of a ($|a|$).
• Example: If graph oscillates between 0.5 and -0.5, amplitude = 0.5.

Period

• Definition: The angle it takes for the function to complete one cycle.
• Calculated using: $\frac{2\pi}{b}$.
• Example Calculation:
  • For $y = a \sin(bx)$, if $b = 4$, then period = $\frac{2\pi}{4} = \frac{\pi}{2}$.

Analyzing Specific Functions

Function: $y = \sin(x)$

• Period:

  • Calculated from the unit circle.
  • Completes one full cycle in $2\pi$ radians.

• Key Features:

  • Midline at y = 0
  • Amplitude: 1
  • Period: $2\pi$
  • Y-values oscillate between 1 and -1.

Function: $y = 3\cos(x)$

• Amplitude:
  • $|3| = 3$
• Period:
  • Remains $2\pi$ (as b = 1).

Function: $y = -\sin(2x)$

• Amplitude:

  • Absolute value of a = 1 (absolute of -1 is 1).
  • Negative sign indicates reflection over x-axis.

• Period:

  • $\frac{2\pi}{2} = \pi$

Frequency

• Definition: The reciprocal of the period.
• Example Calculation:
  • If period = $2\pi$, frequency = $\frac{1}{2\pi}$
  • For $b = \frac{1}{2}$, period = $4\pi$, frequency = $\frac{1}{4\pi}$.
  • Indicates horizontal stretch/shrink of graph.

Summary

• Understanding sine and cosine graphs involves identifying amplitude, period, and the effects of coefficients.
• Amplitude changes with a.
• Period changes with b, calculated as $\frac{2\pi}{b}$.
• Frequency is the reciprocal of the period.

Let me know if you have any questions!