6.1 Graphs of the Sine and Cosine Functions

Sine and cosine functions are the building blocks of periodic behavior. They model repeating patterns in nature, from ocean waves to sound waves. By tweaking their parameters, we can adjust the height, width, and position of these wavy graphs.

Understanding these functions opens up a world of real-life applications. We can predict tides, model Ferris wheel motion, or analyze AC electricity. The key is grasping how each parameter affects the graph's shape and behavior.

Graphs of Sine and Cosine Functions

Variations in sine and cosine graphs

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• General form of sine and cosine functions expressed as $y = a \sin(b(x - c)) + d$ or $y = a \cos(b(x - c)) + d$
  • $a$ represents the amplitude which controls the vertical stretch or compression of the graph
  • $b$ determines the period of the function calculated by $\frac{2\pi}{b}$
    • $b$ is also known as the angular velocity of the function
  • $c$ represents the phase shift which causes a horizontal shift of the graph
  • $d$ represents the vertical shift of the graph up or down
• Amplitude changes affect the height of the graph
  • $|a| > 1$ causes a vertical stretch making the graph taller (mountain range)
  • $0 < |a| < 1$ causes a vertical compression making the graph shorter (ripples on a pond)
  • $a < 0$ reflects the graph across the x-axis flipping it upside down (bat hanging from a cave ceiling)
• Period changes affect the width of one complete cycle of the graph
  • $b > 1$ decreases the period compressing the graph horizontally (high-pitched sound wave)
  • $0 < b < 1$ increases the period stretching the graph horizontally (low-pitched sound wave)
• Phase shift moves the graph horizontally
  • $c > 0$ shifts the graph to the right (delayed start of a pendulum swing)
  • $c < 0$ shifts the graph to the left (early start of a pendulum swing)
• Vertical shift moves the graph up or down
  • $d > 0$ shifts the graph up (raised water level in a tide pool)
  • $d < 0$ shifts the graph down (lowered water level in a tide pool)

Key features of trigonometric functions

• Amplitude $|a|$ measures the maximum vertical distance from the midline to the graph's peaks or troughs (ocean wave height)
• Period $\frac{2\pi}{b}$ represents the length of one complete cycle of the graph (earth's rotation)
  • The frequency of the function is the reciprocal of the period
• Midline is the horizontal line $y = d$ around which the graph oscillates (equilibrium position of a spring)
• Extrema are the maximum and minimum points on the graph
  • For $y = a \sin(b(x - c)) + d$, the maximum is $a + d$ and the minimum is $-a + d$ (temperature fluctuations)
  • For $y = a \cos(b(x - c)) + d$, the maximum is $a + d$ and the minimum is $-a + d$ (voltage in an AC circuit)
• Domain of sine and cosine functions includes all real numbers (infinite input values)
• Range of sine and cosine functions is limited to $[-|a| + d, |a| + d]$ (bounded output values)

Periodic and Sinusoidal Functions

• Sine and cosine functions are examples of periodic functions, which repeat their values at regular intervals
• These functions are also classified as sinusoidal functions due to their characteristic wave-like shape
• The input of these functions can be expressed in degrees or radians, with radians being the preferred unit in advanced mathematics

Transformations for real-world modeling

• Identify the periodic behavior in the real-world situation (tides, seasons, sound waves)
• Determine the key features of the periodic function
  1. Amplitude represents the maximum displacement from the midline (tidal range)
  2. Period represents the time required for one complete cycle (length of a day)
  3. Phase shift represents the horizontal shift of the function (time of high tide)
  4. Vertical shift represents the vertical displacement of the midline (mean sea level)
• Create the appropriate sine or cosine function using the general form $y = a \sin(b(x - c)) + d$ or $y = a \cos(b(x - c)) + d$
• Interpret the graph in the context of the real-world situation
  • Example: modeling the height of a Ferris wheel car over time
    • Amplitude represents the maximum height of the car above the ground (distance from center to top)
    • Period represents the time required for one complete rotation (time for one revolution)
    • Phase shift represents the starting position of the car (angle of car at time zero)
    • Vertical shift represents the height of the center of the Ferris wheel above the ground (distance from ground to center)