You are working on a graphing project in your math class, where you are supposed to graph several functions. Things seem to be going well, until you realize that there is a bold, vertical line three units to the left of where you placed your "y" axis! As it turns out, you've accidentally shifted your entire graph. You didn't notice that your instructor had placed a bold line where the "y" axis was supposed to be. And now, all of the points for your graph of the cosine function are three points farther to the right than they are supposed to be along the "x" axis.

You might be able to keep all of your work, if you can find a way to rewrite the equation so that it takes into account the change in your graph.

Can you think of a way to rewrite the function so that the graph is correct the way you plotted it?

### Horizontal Translations

Horizontal translations involve placing a constant inside the argument of the trig function being plotted. If we return to the example of the parabola,

Here is the graph of

Notice that *adding* 2 to the *to the left*, or in the *negative* direction.

To compare, the graph *to the right* or in the *positive* direction.

We will use the letter **subtracting** **right** and **adding** **left**.

Adding to our previous equations, we now have *opposite sign* of the horizontal shift.

#### Sketch the graph

Sketch

This is a sine wave that has been translated *right*.

Horizontal translations are also referred to as **phase shifts**. Two waves that are identical, but have been moved horizontally are said to be “out of phase” with each other. Remember that cosine and sine are really the same waves with this phase variation.

Alternatively, we could also think of cosine as a sine wave that has been shifted

#### Sketch the graph

Draw a sketch of

This is a cosine curve that has been translated up 1 unit and

#### Graph the function

Graph

This is a sine curve that has been translated 2 units down and moved

Then, take that result and shift it

### Examples

#### Example 1

Earlier, you were asked if you can think of a way to rewrite the function so that the graph is correct the way you plotted it.

As you've now seen by reading this Concept, it is possible to shift an entire graph to the left or the right by changing the argument of the graph.

So in this case, you can keep your graph by changing the function to

#### Example 2

Draw a sketch of

As we've seen, the 3 shifts the graph vertically 3 units, while the

#### Example 3

Draw a sketch of

The

#### Example 4

Draw a sketch of

The 2 added to the function shifts the graph up by 2 units, and the

### Review

Graph each of the following functions.

y=cos(x−π2) y=sin(x+3π2) y=cos(x+π4) y=cos(x−3π4) y=−1+cos(x−π4) y=1+sin(x+π2) y=−2+cos(x+π4) y=3+cos(x−3π2) y=−4+sec(x−π4) y=3+csc(x−π2) y=2+tan(x+π4) y=−3+cot(x−3π2) y=1+cos(x−3π4) - \begin{align*}y=5+\sec(x+\frac{\pi}{2})\end{align*}
- \begin{align*}y=-1+\csc(x+\frac{\pi}{4})\end{align*}
- \begin{align*}y=3+\tan(x-\frac{3\pi}{2})\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.13.