# Translating Sine and Cosine Functions

Graphing Sine and Cosine Functions

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• Horizontal Translation

One way to think about horizontal translations of a function is to think about the value of $x$ that will cause us to find $f \left(0\right)$.

We know the graph of $y = f \left(x\right) = \sin \left(x\right)$.

To graph $y = \sin \left(x - 4\right)$, we can think of it as graphing $y = f \left(x - 4\right)$.
Now, what value of $x$ will make me find $f \left(0\right) = \sin \left(0\right)$? Clearly, it is $x = 4$.
So "$4$ is the new $0$". Everything moves $4$ to the right.

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

To graph $y = \sin \left(x + \frac{\pi}{3}\right)$, we ask ourselves, "What value of $x$ will cause us to find $\sin \left(0\right)$?
That will be $x = - \frac{\pi}{3}$
So, "$- \frac{\pi}{3}$ is the new $0$". Everything moves $- \frac{\pi}{3}$ to the right. Wait a minute, surly it's more clear to say: Everything moves $\frac{\pi}{3}$ to the left.

(For the graph below, remember that $\frac{\pi}{3}$ is a little more than $1$)

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

To start the graph of $y = \sin \left(b x - c\right)$ , we'll need to change the period and also translate.

Cosine

The reasoning for graphing $y = \cos \left(x - h\right)$ is the same. The difference is that
$\sin \left(0\right) = 0$, so the point corresponding to the 'new $0$' goes on the $x$-axis
$\cos 0 = 1$, so the point corresponding to the 'new $0$' goes at $y = 1$

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

• 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!

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