📏Honors Pre-Calculus Unit 6 – 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 graph from the midline to its maximum or minimum point
• Period determines the length of one complete cycle of a periodic function
• Phase shift moves the graph of a periodic function horizontally to the left or right
• Vertical shift moves the graph of a periodic function up or down
• Frequency is the reciprocal of the period and measures the number of cycles per unit of time

Trigonometric Functions Review

• Sine function $(\sin \theta)$ oscillates between -1 and 1 with a period of $2\pi$
• Cosine function $(\cos \theta)$ oscillates between -1 and 1 with a period of $2\pi$, but is shifted $\frac{\pi}{2}$ radians to the left compared to the sine function
• Tangent function $(\tan \theta)$ is the ratio of sine to cosine and has a period of $\pi$
  • Tangent is undefined when $\cos \theta = 0$ (at odd multiples of $\frac{\pi}{2}$)
• Reciprocal trigonometric functions include cosecant $(\csc \theta)$, secant $(\sec \theta)$, and cotangent $(\cot \theta)$
• Trigonometric identities express relationships between trigonometric functions (Pythagorean identity, angle addition formulas)

Periodic Function Basics

• Periodic functions can be represented using the general form $f(x) = A \cdot \sin(B(x - C)) + D$ or $f(x) = A \cdot \cos(B(x - C)) + D$
  • $A$ represents amplitude
  • $B$ is related to period and frequency $(B = \frac{2\pi}{P} = 2\pi f)$
  • $C$ represents phase shift
  • $D$ represents vertical shift
• The domain of a periodic function is all real numbers
• The range of a periodic function depends on its amplitude and vertical shift
• Periodic functions can be even, odd, or neither based on their symmetry properties

Graphing Periodic Functions

• To graph a periodic function, first identify its amplitude, period, phase shift, and vertical shift
• Plot key points such as the maximum, minimum, and zero values of the function
  • For sine and cosine, these points occur at multiples of $\frac{\pi}{2}$ and $\pi$
• Connect the points smoothly to create the graph
• Label the axes with the appropriate scale based on the period and amplitude
• Verify that the graph repeats itself according to the period

Transformations of Periodic Functions

• Amplitude changes affect the height of the graph (stretching or compressing vertically)
  • Multiplying the function by a constant $|k| > 1$ increases amplitude, while $0 < |k| < 1$ decreases amplitude
• Period changes affect the length of one complete cycle (stretching or compressing horizontally)
  • Multiplying the input by a constant $|k| > 1$ decreases period, while $0 < |k| < 1$ increases period
• Phase shift moves the graph horizontally by a constant $h$ (to the right if $h > 0$, to the left if $h < 0$)
• Vertical shift moves the graph up or down by a constant $k$ (up if $k > 0$, down if $k < 0$)
• Combinations of transformations can be applied to create more complex periodic functions

Real-World Applications

• Modeling seasonal temperature variations using sine or cosine functions
  • The average monthly temperature in a city can be approximated by $T(t) = A \cdot \sin(\frac{2\pi}{12}(t - C)) + D$, where $t$ is the month number
• Describing tidal patterns with periodic functions
  • The height of the tide at a given location can be modeled using a sum of sine functions with different amplitudes and periods
• Analyzing sound waves and musical notes
  • The frequency of a musical note determines its pitch, with higher frequencies corresponding to higher pitches
• Studying electrical signals and alternating current (AC)
  • AC voltage can be represented as a sinusoidal function with a specific frequency (e.g., 60 Hz in North America)

Problem-Solving Strategies

• Identify the type of periodic function (sine, cosine, tangent, or their reciprocals)
• Determine the amplitude, period, phase shift, and vertical shift based on the given information
  • Amplitude is often given as the maximum or minimum value of the function
  • Period can be found by identifying the length of one complete cycle or using the frequency
  • Phase shift and vertical shift can be determined by comparing the graph to the parent function
• Write the equation of the periodic function using the general form
• Graph the function or solve for specific values as required by the problem
• Check your solution by verifying that it satisfies the given conditions and makes sense in the context of the problem

Advanced Topics and Extensions

• Damped periodic functions introduce an exponential term to model decay over time (e.g., $f(x) = A e^{-kx} \sin(Bx + C) + D$)
  • The exponential term causes the amplitude to decrease as $x$ increases
• Fourier series represent periodic functions as an infinite sum of sine and cosine terms with different frequencies and amplitudes
  • Fourier series are used in signal processing, heat transfer, and other applications
• Lissajous curves are graphs produced by combining two perpendicular harmonic motions with different frequencies
  • These curves can be used to study the relationship between frequency ratios and graph shapes
• Polar equations can be used to represent periodic functions in polar form (e.g., $r = A \sin(B\theta + C) + D$)
  • Polar periodic functions create interesting symmetrical patterns when graphed
• Periodic functions can be used in calculus to study rates of change, accumulation, and optimization problems involving cyclic phenomena