Periodic Functions: Cheat Sheet

Essential Concepts

Graphs of Sine and Cosine Functions

  • Key Features
    • Period [latex]2\pi[/latex]
    • Domain: [latex](-\infty, \infty)[/latex]
    • Range: [latex][-1, 1][/latex]
    • [latex]y = \sin x[/latex] is an odd function (symmetric about the origin)
    • [latex]y = \cos x[/latex] is an even function (symmetric about the y-axis)
  • General form equation
    • [latex]y = A\sin(Bx - C) + D[/latex]
    • [latex]y = A\cos(Bx - C) + D[/latex]
  • Transformations:
    • Amplitude: [latex]|A|[/latex] measures the vertical stretch/compression
    • Period: [latex]P = \frac{2\pi}{|B|}[/latex] is the length of one complete cycle
      • If [latex]|B| > 1[/latex]: horizontal compression (period less than [latex]2\pi[/latex])
      • If [latex]|B| < 1[/latex]: horizontal stretch (period greater than [latex]2\pi[/latex])
    • Phase shift: [latex]\frac{C}{B}[/latex] represents horizontal displacement
      • If [latex]C > 0[/latex]: shift right
      • If [latex]C < 0[/latex]: shift left
    • Vertical shift: [latex]D[/latex] is the midline of the graph
      • Positive [latex]D[/latex]: shift up
      • Negative [latex]D[/latex]: shift down
    • Reflection: If [latex]A < 0[/latex], the graph is reflected across the x-axis

Graphing sine and cosine:

  1. Identify amplitude [latex]|A|[/latex], period [latex]\frac{2\pi}{|B|}[/latex], phase shift [latex]\frac{C}{B}[/latex], and midline [latex]y = D[/latex]
  2. Mark key points at quarter-period intervals
  3. Apply transformations in order: horizontal stretch/compression, horizontal shift, vertical stretch/compression, vertical shift, reflection

Graphs of the Other Trigonometric Functions

Tangent function [latex]y = A\tan(Bx - C) + D[/latex]:

  • Period: [latex]P = \frac{\pi}{|B|}[/latex]
  • Domain: [latex]x \ne \frac{C}{B} + \frac{\pi}{2|B|}k[/latex], where [latex]k[/latex] is an odd integer
  • Range: [latex](-\infty, \infty)[/latex]
  • Vertical asymptotes: [latex]x = \frac{C}{B} + \frac{\pi}{2|B|}k[/latex], where [latex]k[/latex] is an odd integer
  • No amplitude; [latex]|A|[/latex] is the stretching/compressing factor
  • Odd function

Cotangent function [latex]y = A\cot(Bx - C) + D[/latex]:

  • Period: [latex]P = \frac{\pi}{|B|}[/latex]
  • Domain: [latex]x \ne \frac{C}{B} + \frac{\pi}{|B|}k[/latex], where [latex]k[/latex] is an integer
  • Range: [latex](-\infty, \infty)[/latex]
  • Vertical asymptotes: [latex]x = \frac{C}{B} + \frac{\pi}{|B|}k[/latex], where [latex]k[/latex] is an integer
  • No amplitude; [latex]|A|[/latex] is the stretching/compressing factor
  • Odd function

Secant function [latex]y = A\sec(Bx - C) + D[/latex]:

  • Period: [latex]P = \frac{2\pi}{|B|}[/latex]
  • Domain: [latex]x \ne \frac{C}{B} + \frac{\pi}{2|B|}k[/latex], where [latex]k[/latex] is an odd integer
  • Range: [latex](-\infty, -|A| + D] \cup [|A| + D, \infty)[/latex]
  • Vertical asymptotes: [latex]x = \frac{C}{B} + \frac{\pi}{2|B|}k[/latex], where [latex]k[/latex] is an odd integer
  • Even function
  • Graph using reciprocal relationship: [latex]\sec x = \frac{1}{\cos x}[/latex]

Cosecant function [latex]y = A\csc(Bx - C) + D[/latex]:

  • Period: [latex]P = \frac{2\pi}{|B|}[/latex]
  • Domain: [latex]x \ne \frac{C}{B} + \frac{\pi}{|B|}k[/latex], where [latex]k[/latex] is an integer
  • Range: [latex](-\infty, -|A| + D] \cup [|A| + D, \infty)[/latex]
  • Vertical asymptotes: [latex]x = \frac{C}{B} + \frac{\pi}{|B|}k[/latex], where [latex]k[/latex] is an integer
  • Odd function
  • Graph using reciprocal relationship: [latex]\csc x = \frac{1}{\sin x}[/latex]

Graphing tangent and cotangent:

  1. Identify stretching factor [latex]|A|[/latex] and period [latex]\frac{\pi}{|B|}[/latex]
  2. Locate vertical asymptotes
  3. Plot reference points including the center point and quarter-period points
  4. Draw the curve approaching asymptotes correctly

Graphing secant and cosecant:

  1. First sketch the corresponding cosine or sine function
  2. Use reciprocal relationship to draw secant or cosecant
  3. Vertical asymptotes occur where cosine or sine equals zero
  4. Local extrema of secant/cosecant occur at extrema of cosine/sine

Key Equations

General sine function [latex]y = A\sin(Bx - C) + D[/latex]
General cosine function [latex]y = A\cos(Bx - C) + D[/latex]
Period of sine/cosine [latex]P = \frac{2\pi}{|B|}[/latex]
Phase shift [latex]\frac{C}{B}[/latex]
Midline [latex]y = D[/latex]
General tangent function [latex]y = A\tan(Bx - C) + D[/latex]
Tangent asymptotes [latex]x = \frac{C}{B} + \frac{\pi}{2|B|}k[/latex], [latex]k[/latex] odd integer
General cotangent function [latex]y = A\cot(Bx - C) + D[/latex]
Cotangent asymptotes [latex]x = \frac{C}{B} + \frac{\pi}{|B|}k[/latex], [latex]k[/latex] integer
Period of tangent/cotangent [latex]P = \frac{\pi}{|B|}[/latex]
General secant function [latex]y = A\sec(Bx - C) + D[/latex]
General cosecant function [latex]y = A\csc(Bx - C) + D[/latex]
Period of secant/cosecant [latex]P = \frac{2\pi}{|B|}[/latex]

Glossary

amplitude

The distance from the midline to the maximum or minimum of a sinusoidal function; calculated as [latex]|A|[/latex] in the general forms [latex]y = A\sin(Bx - C) + D[/latex] or [latex]y = A\cos(Bx - C) + D[/latex]

cosecant function

The function [latex]\csc x = \frac{1}{\sin x}[/latex], which has period [latex]2\pi[/latex], vertical asymptotes where [latex]\sin x = 0[/latex], and is undefined at those points

cotangent function

The function [latex]\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}[/latex], which has period [latex]\pi[/latex], vertical asymptotes where [latex]\sin x = 0[/latex], and range [latex](-\infty, \infty)[/latex]

midline

The horizontal line [latex]y = D[/latex] that runs through the middle of a sinusoidal graph, located halfway between the maximum and minimum values

period of a function

The length of one complete cycle of a periodic function’s graph; for sine and cosine, [latex]P = \frac{2\pi}{|B|}[/latex]; for tangent and cotangent, [latex]P = \frac{\pi}{|B|}[/latex]

phase shift

The horizontal displacement of a periodic function from its standard position; calculated as [latex]\frac{C}{B}[/latex] in the general form [latex]y = A\sin(Bx - C) + D[/latex] or similar

reciprocal identity

An identity expressing one trigonometric function as the reciprocal of another, such as [latex]\sec x = \frac{1}{\cos x}[/latex] or [latex]\csc x = \frac{1}{\sin x}[/latex]

secant function

The function [latex]\sec x = \frac{1}{\cos x}[/latex], which has period [latex]2\pi[/latex], vertical asymptotes where [latex]\cos x = 0[/latex], and is undefined at those points

sinusoidal function

A function that has the same general shape as a sine or cosine function

stretching/compressing factor

The coefficient [latex]|A|[/latex] in trigonometric functions that do not have amplitude (tangent, cotangent, secant, cosecant); indicates vertical stretch or compression

tangent function

The function [latex]\tan x = \frac{\sin x}{\cos x}[/latex], which has period [latex]\pi[/latex], vertical asymptotes where [latex]\cos x = 0[/latex], and range [latex](-\infty, \infty)[/latex]

vertical shift

The vertical displacement [latex]D[/latex] that moves the midline of a periodic function up or down from [latex]y = 0[/latex]