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

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

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Answer 1 · 2015-04-12T08:15:52

For an equation:

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

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

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

A horizontal translation is of the form:
$y = sin (θ + A)$ $y = sin(theta + A)$ where $A \neq 0$ $A!=0$

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

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

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

and

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

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

That is, the original graph of $y = cos θ$ $y= costheta$ 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!

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

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