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

2 Answers
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=sinx+ky=sinx+k Which is the same as y=k+sinxy=k+sinx:

In this case we start with a number (or angle) xx. We find the sine of xx, which will be a number between -11 and 11. The after that, we get yy by adding kk. (Remember that kk could be negative.)

This gives us a final yy value betwee -1+k1+k and 1+k1+k.

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

Examples:

y=sinxy=sinx
graph{y=sinx [-5.578, 5.52, -1.46, 4.09]}

y=sinx+2 = 2+sinxy=sinx+2=2+sinx
graph{y=sinx+2 [-5.578, 5.52, -1.46, 4.09]}

y=sinx-4=-4+sinxy=sinx4=4+sinx
graph{y=sinx-4 [-5.58, 5.52, -5.17, 0.38]}

The reasoning is the same for y=cosx+k=k+cosxy=cosx+k=k+cosx, but the starting graph looks different, so the final graph is also different:

y=cosxy=cosx
graph{y=cosx [-5.578, 5.52, -1.46, 4.09]}

y=cosx+2 = 2+cosxy=cosx+2=2+cosx
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 xx that will cause us to find f(0)f(0).

We know the graph of y=f(x)=sin(x)y=f(x)=sin(x).

To graph y=sin(x-4)y=sin(x4), we can think of it as graphing y=f(x-4)y=f(x4).
Now, what value of xx will make me find f(0)=sin(0)f(0)=sin(0)? Clearly, it is x=4x=4.
So "44 is the new 00". Everything moves 44 to the right.

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

To graph y=sin(x+ pi/3)y=sin(x+π3), we ask ourselves, "What value of xx will cause us to find sin(0)sin(0)?
That will be x=- pi/3x=π3
So, "- pi/3π3 is the new 00". Everything moves - pi/3π3 to the right. Wait a minute, surly it's more clear to say: Everything moves pi/3π3 to the left.

(For the graph below, remember that pi/3π3 is a little more than 11)

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

To start the graph of y=sin(bx- c)y=sin(bxc) , we'll need to change the period and also translate.

Cosine

The reasoning for graphing y=cos(x-h)y=cos(xh) is the same. The difference is that
sin(0)=0sin(0)=0, so the point corresponding to the 'new 00' goes on the xx-axis
cos0=1cos0=1, so the point corresponding to the 'new 00' goes at y=1y=1

y = cos (x+ pi/3)y=cos(x+π3)
graph{y=cos(x+pi/3) [-3.02, 1.845, -1.192, 1.241]}