Translations of trig graphs

Translations of Trig Graphs

Basic Understandings

• Trig graphs pertain to graphs of trigonometric functions like sine, cosine and tangent.
• A basic understanding of what these functions generate on a graph is vital. For instance, the graph of sine and cosine gives a wave, while tangent gives an infinite series of vertical asymptotes.

Describing a Translation

• A translation refers to a shift of the graph either along the x-axis or the y-axis, or both.
• A translation can be represented by a vector, written as (x, y), where ‘x’ denotes the shift along the x-axis (horizontal) while ‘y’ is the shift along the y-axis (vertical).

Translating Sine and Cosine Graphs

• The graph of y = sin x or y = cos x is a wave that repeats every 2π along the x-axis.
• If the graph is translated horizontally by adding a value ‘h’ to ‘x’, this affects the phase of the trig function (y = sin (x + h) or y = cos (x + h)).
• If a positive ‘h’ value is added, the graph moves left by ‘h’ units, while a negative ‘h’ value shifts the graph to the right ‘h’ units.
• Vertical translation involves adding or subtracting a number ‘v’ from ‘y’. For instance, for y = sin x, if ‘v’ is added, we get y = sin x + v.
• A positive ‘v’ value moves the graph up ‘v’ units, while a negative ‘v’ value moves it down ‘v’ units.

Translating Tangent Graphs

• The graph of y = tan x displays vertical asymptotes and repeats every π.
• The above explained principles of translation can also be applied to this graph, with the exception that the period of repetition is different from sine and cosine functions.

Understanding Translations of Trig Graphs including how to execute and interpret these translations is a fundamental component of Geometry. Always remember the described rules and engage in ample practice to deepen this understanding.