What if you had the square root function ? How would the graph of the function change if you added 5 to the righthand side of the equation or if you multiplied *x* by 3? After completing this Concept, you'll be able to identify various shifts in square root functions.

### Watch This

CK-12 Foundation: Shifts of Square Root Functions

### Guidance

We will now look at how graphs are shifted up and down in the Cartesian plane.

#### Example A

*Graph the functions* *and* .

**Solution**

When we add a constant to the right-hand side of the equation, the graph keeps the same shape, but shifts up for a positive constant or down for a negative one.

#### Example B

*Graph the functions* *and* .

**Solution**

When we add a constant to the **argument** of the function (the part under the radical sign), the function shifts to the left for a positive constant and to the right for a negative constant.

Now let’s see how to combine all of the above types of transformations.

#### Example C

*Graph the function* .

**Solution**

We can think of this function as a combination of shifts and stretches of the basic square root function . We know that the graph of that function looks like this:

If we multiply the argument by 3 to obtain , this stretches the curve vertically because the value of increases faster by a factor of .

Next, when we subtract 1 from the argument to obtain this shifts the entire graph to the left by one unit.

Multiplying the function by a factor of 2 to obtain stretches the curve vertically again, because increases faster by a factor of 2.

Finally we add 2 to the function to obtain . This shifts the entire function vertically by 2 units.

Each step of this process is shown in the graph below. The purple line shows the final result.

Now we know how to graph square root functions without making a table of values. If we know what the basic function looks like, we can use shifts and stretches to **transform** the function and get to the desired result.

Watch this video for help with the Examples above.

CK-12 Foundation: Shifts of Square Root Functions

### Vocabulary

- For the
**square root function**with the form: , is the vertical shift.

### Guided Practice

*Graph the function* .

**Solution**

We can think of this function as a combination of shifts and stretches of the basic square root function . We know that the graph of that function looks like this:

Next, when we add 3 to the argument to obtain this shifts the entire graph to the right by 3 units.

Multiplying the function by -1 to obtain which reflects the function across the -axis.

Finally we subtract 5 from the function to obtain . This shifts the entire function down vertically by 5 units.

### Practice

Graph the following functions.