# How do you identify the vertical and horizontal translations of sine and cosine from a graph and an equation?

Apr 12, 2015

For an equation:

A vertical translation is of the form:
$y = \sin \left(\theta\right) + A$ where $A \ne 0$
OR $y = \cos \left(\theta\right) + A$

Example: $y = \sin \left(\theta\right) + 5$ is a $\sin$ graph that has been shifted up by 5 units

The graph $y = \cos \left(\theta\right) - 1$ is a graph of $\cos$ shifted down the y-axis by 1 unit

A horizontal translation is of the form:
$y = \sin \left(\theta + A\right)$ where $A \ne 0$

Examples:
The graph $y = \sin \left(\theta + \frac{\pi}{2}\right)$ is a graph of $\sin$ that has been shifted $\frac{\pi}{2}$ radians to the right

For a graph:
I'm to illustrate with an example given above:

For compare:
$y = \cos \left(\theta\right)$
graph{cosx [-5.325, 6.675, -5.16, 4.84]}

and

$y = \cos \left(\theta\right) - 1$
graph{cosx -1 [-5.325, 6.675, -5.16, 4.84]}
To verify that the graph of $y = \cos \left(\theta\right) - 1$ is a vertical translation, if you look on the graph,

the point where $\theta = 0$ is no more at $y = 1$ it is now at $y = 0$

That is, the original graph of $y = \cos \theta$ has been shifted down by 1 unit.

Another way to look at it is to see that, every point has been brought down 1 unit!