Question

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

Answer 1

For an equation:

A vertical translation is of the form:
`y = sin(theta) + A` where `A!=0`
OR `y = cos(theta) + A`

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

The graph `y = cos(theta) - 1` is a graph of `cos` shifted down the y-axis by 1 unit

A horizontal translation is of the form:
`y = sin(theta + A)` where `A!=0`

Examples:
The graph `y = sin(theta + pi/2)` is a graph of `sin` that has been shifted `pi/2` radians to the right

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

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

and

`y = cos(theta) - 1`
graph{cosx -1 [-5.325, 6.675, -5.16, 4.84]}
To verify that the graph of `y = cos(theta) - 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= costheta` 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!