# Shift a Sine Function in a Graph

Playing around with the amplitude and period of the sine curve can result in some interesting changes to the basic curve on a graph. That curve is still recognizable, though. You can see the rolling, smooth curve crossing back and forth over a middle line.

In addition to those changes, you have two other options for altering the sine curve — shifting the curve up or down, or sideways. These shifts are called *translations* of the curve.

## Sliding a function up or down on a graph

You can move a sine curve up or down by simply adding or subtracting a number from the equation of the curve. For example, the graph of *y* = sin *x* + 4 moves the whole curve up 4 units, with the sine curve crossing back and forth over the line *y* = 4. On the other hand, the graph of *y* = sin *x* – 1 slides everything down 1 unit. The following figure shows what the two graphs look like.

*y*= sin

*x*+ 4 and

*y*= sin

*x*– 1.

As you can see, the basic shape of the sine curve is still recognizable — the curves are just shifted up or down on the coordinate plane.

## Sliding a function left or right on a graph

By adding or subtracting a number from the angle (variable) in a sine equation, you can move the curve to the left or right of its usual position. A shift, or translation, of 90 degrees can change the sine curve to the cosine curve. But the translation of the sine itself is important: Shifting the curve left or right can change the places that the curve crosses the *x*-axis or some other horizontal line.

For example, the graph of *y* = sin (*x* + 1) results in the usual sine curve slid 1 unit to the left, and the graph of *y* = sin (*x* – 3) slides it 3 units to the right. The below figure shows the graphs of the original sine equation and these two shifted equations.

*y*= sin

*x*,

*y*= sin (

*x*+ 1), and

*y*= sin (

*x*– 3).

Take a look at the point marked on each graph in the above figure. This point illustrates how an *intercept* (where the curve crosses an axis) shifts on the graph when you add or subtract a number from the angle variable.

Note the difference between adding or subtracting a number to the function and adding or subtracting a number to the angle measure. These operations affect the curve differently, as you can see by comparing the preceding figures.

*y* = sin *x* + 2: Adding 2 to the function raises the curve by 2 units.

*y* = sin (*x* + 2): Adding 2 to the angle variable shifts the curve 2 units to the left.