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

Mar 22, 2015

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

Vertical translation

Graphing $y = \sin x + k$ Which is the same as $y = k + \sin x$:

In this case we start with a number (or angle) $x$. We find the sine of $x$, which will be a number between $- 1$ and $1$. The after that, we get $y$ by adding $k$. (Remember that $k$ could be negative.)

This gives us a final $y$ value betwee $- 1 + k$ and $1 + k$.

This will translate the graph up if $k$ is positive ($k > 0$)
or down if $k$ is negative ($k < 0$)

Examples:

$y = \sin x$
graph{y=sinx [-5.578, 5.52, -1.46, 4.09]}

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

$y = \sin x - 4 = - 4 + \sin x$
graph{y=sinx-4 [-5.58, 5.52, -5.17, 0.38]}

The reasoning is the same for $y = \cos x + k = k + \cos x$, but the starting graph looks different, so the final graph is also different:

$y = \cos x$
graph{y=cosx [-5.578, 5.52, -1.46, 4.09]}

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

Mar 22, 2015

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]}