What if you had the square root function \begin{align*}y=\sqrt{x}\end{align*}? 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* \begin{align*}y=\sqrt{x}, y=\sqrt{x} + 2\end{align*} *and* \begin{align*}y=\sqrt{x} - 2\end{align*}.

**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* \begin{align*}y=\sqrt{x}, y=\sqrt{x - 2},\end{align*} *and* \begin{align*}y = \sqrt{x + 2}\end{align*}.

**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* \begin{align*}y = 2\sqrt{3x - 1} + 2\end{align*}.

**Solution**

We can think of this function as a combination of shifts and stretches of the basic square root function \begin{align*}y = \sqrt{x}\end{align*}. We know that the graph of that function looks like this:

If we multiply the argument by 3 to obtain \begin{align*}y = \sqrt{3x}\end{align*}, this stretches the curve vertically because the value of \begin{align*}y\end{align*} increases faster by a factor of \begin{align*}\sqrt{3}\end{align*}.

Next, when we subtract 1 from the argument to obtain \begin{align*}y = \sqrt{3x - 1}\end{align*} this shifts the entire graph to the left by one unit.

Multiplying the function by a factor of 2 to obtain \begin{align*}y = 2 \sqrt{3x - 1}\end{align*} stretches the curve vertically again, because \begin{align*}y\end{align*} increases faster by a factor of 2.

Finally we add 2 to the function to obtain \begin{align*}y = 2 \sqrt{3x - 1} + 2\end{align*}. 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: \begin{align*}y = a \sqrt{f(x)} + c\end{align*}, \begin{align*}c\end{align*} is the vertical shift.

### Guided Practice

*Graph the function* \begin{align*}y = -\sqrt{x +3} -5\end{align*}.

**Solution**

We can think of this function as a combination of shifts and stretches of the basic square root function \begin{align*}y = \sqrt{x}\end{align*}. We know that the graph of that function looks like this:

Next, when we add 3 to the argument to obtain \begin{align*}y = \sqrt{x +3}\end{align*} this shifts the entire graph to the right by 3 units.

Multiplying the function by -1 to obtain \begin{align*}y = - \sqrt{x +3}\end{align*} which reflects the function across the \begin{align*}x\end{align*}-axis.

Finally we subtract 5 from the function to obtain \begin{align*}y = - \sqrt{x +3}-5\end{align*}. This shifts the entire function down vertically by 5 units.

### Practice

Graph the following functions.

- \begin{align*}y = \sqrt{2x - 1}\end{align*}
- \begin{align*}y = \sqrt{x - 100}\end{align*}
- \begin{align*}y = \sqrt{4x + 4}\end{align*}
- \begin{align*}y = \sqrt{5 - x}\end{align*}
- \begin{align*}y = 2\sqrt{x} + 5\end{align*}
- \begin{align*}y = 3 - \sqrt{x}\end{align*}
- \begin{align*}y = 4 + 2 \sqrt{x}\end{align*}
- \begin{align*}y = 2 \sqrt{2x + 3} + 1\end{align*}
- \begin{align*}y = 4 + \sqrt{2 - x}\end{align*}
- \begin{align*}y = \sqrt{x + 1} - \sqrt{4x - 5}\end{align*}