What if you had the square root function *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

If we multiply the argument by 3 to obtain

Next, when we subtract 1 from the argument to obtain

Multiplying the function by a factor of 2 to obtain

Finally we add 2 to the function to obtain

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

### 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

Next, when we add 3 to the argument to obtain

Multiplying the function by -1 to obtain

Finally we subtract 5 from the function to obtain

### Explore More

Graph the following functions.

y=2x−1−−−−−√ y=x−100−−−−−−√ y=4x+4−−−−−√ y=5−x−−−−√ y=2x√+5 y=3−x√ y=4+2x√ y=22x+3−−−−−√+1 y=4+2−x−−−−√ y=x+1−−−−√−4x−5−−−−−√

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 11.2.