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# Shifts of Square Root Functions

## Translate square root functions vertically and horizontally

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Shifts of Square Root Functions

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.

### 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.

### 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.

1. \begin{align*}y = \sqrt{2x - 1}\end{align*}
2. \begin{align*}y = \sqrt{x - 100}\end{align*}
3. \begin{align*}y = \sqrt{4x + 4}\end{align*}
4. \begin{align*}y = \sqrt{5 - x}\end{align*}
5. \begin{align*}y = 2\sqrt{x} + 5\end{align*}
6. \begin{align*}y = 3 - \sqrt{x}\end{align*}
7. \begin{align*}y = 4 + 2 \sqrt{x}\end{align*}
8. \begin{align*}y = 2 \sqrt{2x + 3} + 1\end{align*}
9. \begin{align*}y = 4 + \sqrt{2 - x}\end{align*}
10. \begin{align*}y = \sqrt{x + 1} - \sqrt{4x - 5}\end{align*}

### Vocabulary Language: English

square root function

square root function

A square root function is a function with the parent function $y=\sqrt{x}$.
Transformations

Transformations

Transformations are used to change the graph of a parent function into the graph of a more complex function.