Translations of sine & Cosine Graphs

How the equation relates to the graphs

A translation is a type of transformation that is isometric (isometric means that the shape is not distorted in any way). A translation of a graph, whether its sine or cosine or anything, can be thought of a 'slide'. To translate a graph, all that you have to do is shift or slide the entire graph to a different place.

Vertical Translations of Sine and Cosine graphs

Example

If you're on this web page, you should be very familiar with the graph of y =sin(x) as shown below $$ 0 \le x \le 2 \pi $$ . An example of first type of translation that we wil look at is y = sin(x) +1

Below you can see both the original graph of y =sin(x) and the graph of the translation y = sin(x) + 1

translation of the graph of sinx downward by one
Problem 1

Based on the example above you can figure out, what the graph of the following translation would look like y = sin(x)− 1

This translation expresses a vertical shift downwards by 1.

Problem 2

What would the graph of the following equation look like: $$ \frac{1}{2} sin(2x) - \frac{1}{2} $$ ?
(Note this question assumes that you have studied amplitude of sine equationsas well as period of sine and cosine graphs)