Trigonometric Graphs

Trigonometric graphs represent the periodic behavior of sine, cosine, and tangent functions, along with their transformations. These graphs are widely used to model real-life phenomena.

Key Concepts

• Circular Functions: The standard forms of sine (\(\sin x\)), cosine (\(\cos x\)), and tangent (\(\tan x\)) functions.

• Properties of Trigonometric Functions:
  • Amplitude: The height of the wave from the midline.
  • Period: The length of one complete cycle.
  • Phase Shift: Horizontal shift of the graph.
  • Vertical Shift: Movement of the graph up or down.
• Composite Functions: Forms like \(f(x) = a\sin(b(x + c)) + d\) and their components:
  • \(a\): Amplitude (stretch/compression in the \(y\)-direction).
  • \(b\): Controls the period (\(\text{Period} = \frac{2\pi}{b}\)).
  • \(c\): Phase shift (horizontal shift by \(-c\)).
  • \(d\): Vertical shift (translation along \(y\)-axis).
• Transformations: Graph modifications such as:
  • Stretch/compression in \(y\)-direction by multiplying \(y\)-values by a factor.
  • Stretch/compression in \(x\)-direction by multiplying \(x\)-values by the reciprocal of a factor.
  • Translation in \(x\)- or \(y\)-direction by adding/subtracting values to/from \(x\) or \(y\).

Examples

Example : Graphing Composite Functions

• Graph \(f(x) = 2\cos(3(x – \frac{\pi}{4})) + 1\):
  • \(a = 2\): Amplitude is 2.
  • \(b = 3\): Period is \(\frac{2\pi}{3}\).
  • \(c = -\frac{\pi}{4}\): Phase shift is \(\frac{\pi}{4}\) to the right.
  • \(d = 1\): Vertical shift is 1 unit up.