# Translations of Trigonometric Functions

## Understanding Trigonometric Function Translations

When graphed, trigonometric functions often undergo transformations, resulting in a shift on the coordinate plane. These shifts, known as translations, can occur horizontally or vertically. In this article, we will explore the different types of trigonometric function translations and the specific rules for each, using real-life examples.

## Types of Trigonometric Function Translations

Trigonometric function translations can be categorized into two main types: horizontal and vertical. Horizontal translations involve shifting the graph left or right, while vertical translations result in a movement up or down on the coordinate plane. Let's take a closer look at each type and how to perform them.

### Horizontal Translations

If a trigonometric function is written in the form f(x) = sin(ax), where a is a constant, then the graph of sin(x) will shift a units left or right depending on the sign of a. This is also known as a phase shift. There are two possible scenarios:

• If a is negative, the graph will shift to the left. See the graph below, where the dashed green line represents the newly shifted graph after adding a inside the parentheses:

Example of a horizontal translation with a negative coefficient:

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