# How do you graph sine and cosine functions when it is translated?

##### 2 Answers

I think you'll find a useful answer here: http://socratic.org/trigonometry/graphing-trigonometric-functions/translating-sine-and-cosine-functions

**Vertical translation**

Graphing

In this case we start with a number (or angle)

This gives us a final

This will translate the graph up if

or down if

Examples:

graph{y=sinx [-5.578, 5.52, -1.46, 4.09]}

graph{y=sinx+2 [-5.578, 5.52, -1.46, 4.09]}

graph{y=sinx-4 [-5.58, 5.52, -5.17, 0.38]}

The reasoning is the same for

graph{y=cosx [-5.578, 5.52, -1.46, 4.09]}

graph{y=cosx+2 [-5.578, 5.52, -1.46, 4.09]}

**Horizontal Translation**

One way to think about horizontal translations of a function is to think about the value of

We know the graph of

To graph

Now, what value of

So "

graph{y=sin(x-4) [-0.498, 7.295, -2.302, 1.596]}

To graph

That will be

So, "

(For the graph below, remember that

graph{y=sin(x+pi/3) [-3.02, 1.845, -1.192, 1.241]}

To start the graph of

**Cosine**

The reasoning for graphing

graph{y=cos(x+pi/3) [-3.02, 1.845, -1.192, 1.241]}