📈College Algebra Unit 8 – Periodic Functions

Key Concepts

• Periodic functions repeat their values at regular intervals called periods
• Trigonometric functions (sine, cosine, tangent) are common examples of periodic functions
• Amplitude measures the height of a periodic function's peaks and troughs from its midline
• Period determines the length of one complete cycle of a periodic function
• Phase shift moves a periodic function horizontally to the left or right
• Vertical shift moves a periodic function up or down
• Frequency is the reciprocal of the period and measures the number of cycles per unit of time
• Midline is the horizontal line that runs through the center of a periodic function's graph

Trigonometric Functions

• Sine function $(\sin x)$ oscillates between -1 and 1 with a period of $2\pi$
  • Starts at the origin, reaches a maximum at $\frac{\pi}{2}$, and a minimum at $\frac{3\pi}{2}$
• Cosine function $(\cos x)$ also oscillates between -1 and 1 with a period of $2\pi$
  • Starts at 1, reaches a minimum at $\pi$, and a maximum at $2\pi$
• Tangent function $(\tan x)$ has a period of $\pi$ and undefined values at odd multiples of $\frac{\pi}{2}$
  • Oscillates between positive and negative infinity
• Cosecant $(\csc x)$, secant $(\sec x)$, and cotangent $(\cot x)$ are reciprocals of sine, cosine, and tangent, respectively
• Trigonometric functions can be transformed using amplitude, period, phase shift, and vertical shift

Graphing Periodic Functions

• Start by graphing the basic function without any transformations
• Apply amplitude changes by multiplying the function by a constant $|a|$
  • If $|a| > 1$, the graph is stretched vertically; if $0 < |a| < 1$, the graph is compressed vertically
• Modify the period by dividing the input variable by a constant $|b|$
  • The new period is $\frac{2\pi}{|b|}$; if $|b| > 1$, the period decreases, and if $0 < |b| < 1$, the period increases
• Shift the graph horizontally by adding or subtracting a constant $c$ inside the function
  • Positive $c$ shifts the graph to the left, while negative $c$ shifts it to the right
• Shift the graph vertically by adding or subtracting a constant $d$ outside the function
  • Positive $d$ shifts the graph up, while negative $d$ shifts it down

Properties and Characteristics

• Periodic functions have a constant period, amplitude, and midline
• Even functions are symmetric about the y-axis $(f(-x) = f(x))$, while odd functions are symmetric about the origin $(f(-x) = -f(x))$
• Trigonometric functions have specific even/odd properties (sine is odd, cosine is even, tangent is odd)
• Periodic functions can be continuous or discontinuous
  • Continuous functions have no breaks or gaps in their graphs
  • Discontinuous functions have breaks, gaps, or undefined points (tangent, secant, cosecant)
• Periodic functions can be bounded or unbounded
  • Bounded functions have a maximum and minimum value (sine, cosine)
  • Unbounded functions have no maximum or minimum value (tangent, secant, cosecant)

Real-World Applications

• Modeling seasonal changes in temperature, daylight hours, or animal populations
• Describing the motion of pendulums, springs, or waves (sound, light, water)
• Analyzing alternating current (AC) in electrical systems
• Studying the behavior of tides, planetary orbits, or celestial bodies
• Representing periodic trends in economics, such as business cycles or stock market fluctuations
• Modeling the vibrations of strings in musical instruments
• Describing the motion of pistons in engines or the rotation of gears in machinery

Problem-Solving Strategies

• Identify the type of periodic function (sine, cosine, tangent, or their reciprocals)
• Determine the basic function's period, amplitude, and midline
• Identify any transformations applied to the basic function
  • Changes in amplitude, period, phase shift, or vertical shift
• Write the equation of the transformed function using the general form:
  • $a \cdot f(b(x - c)) + d$, where $f$ is the basic function
• Graph the transformed function by applying the transformations to the basic function's graph
• Analyze the graph to find key points, such as maxima, minima, or zeros
• Use the graph or equation to answer questions about the function's properties or behavior

Common Mistakes and Misconceptions

• Confusing the effects of changes in amplitude, period, phase shift, and vertical shift
  • Remember: amplitude affects height, period affects width, phase shift moves horizontally, vertical shift moves vertically
• Incorrectly applying transformations to the basic function
  • Changes inside the function (period and phase shift) are applied before changes outside (amplitude and vertical shift)
• Misinterpreting the period as the frequency or vice versa
  • Period is the length of one complete cycle, while frequency is the number of cycles per unit of time
• Forgetting to consider the domain restrictions of certain functions (tangent, secant, cosecant)
• Mistaking even functions for odd functions or vice versa
  • Even functions are symmetric about the y-axis, while odd functions are symmetric about the origin
• Incorrectly identifying the midline of a transformed function
  • The midline is affected by the vertical shift, not the amplitude

Advanced Topics

• Combining periodic functions through addition, subtraction, multiplication, or composition
  • The resulting function may have a different period, amplitude, or shape than the original functions
• Analyzing the Fourier series representation of periodic functions
  • Any periodic function can be expressed as an infinite sum of sine and cosine functions
• Studying the relationship between trigonometric functions and the unit circle
  • The sine and cosine of an angle can be represented as the y and x coordinates of a point on the unit circle
• Exploring the connections between periodic functions and complex numbers (Euler's formula)
  • Euler's formula relates exponential functions with complex arguments to trigonometric functions
• Investigating the use of periodic functions in differential equations and harmonic motion
• Applying periodic functions to signal processing, Fourier transforms, and frequency analysis
• Examining the role of periodic functions in physics, such as wave mechanics and quantum mechanics