11.2: Shifts of Square Root Functions
What if you had the square root function \begin{align*}y=\sqrt{x}\end{align*}
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CK12 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*}
Solution
When we add a constant to the righthand 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*}
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*}
If we multiply the argument by 3 to obtain \begin{align*}y = \sqrt{3x}\end{align*}
Next, when we subtract 1 from the argument to obtain \begin{align*}y = \sqrt{3x  1}\end{align*}
Multiplying the function by a factor of 2 to obtain \begin{align*}y = 2 \sqrt{3x  1}\end{align*}
Finally we add 2 to the function to obtain \begin{align*}y = 2 \sqrt{3x  1} + 2\end{align*}
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.
CK12 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*}
y=af(x)−−−−√+c , \begin{align*}c\end{align*}c 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*}
Next, when we add 3 to the argument to obtain \begin{align*}y = \sqrt{x +3}\end{align*}
Multiplying the function by 1 to obtain \begin{align*}y =  \sqrt{x +3}\end{align*}
Finally we subtract 5 from the function to obtain \begin{align*}y =  \sqrt{x +3}5\end{align*}
Practice
Graph the following functions.

\begin{align*}y = \sqrt{2x  1}\end{align*}
y=2x−1−−−−−√ 
\begin{align*}y = \sqrt{x  100}\end{align*}
y=x−100−−−−−−√ 
\begin{align*}y = \sqrt{4x + 4}\end{align*}
y=4x+4−−−−−√ 
\begin{align*}y = \sqrt{5  x}\end{align*}
y=5−x−−−−−√ 
\begin{align*}y = 2\sqrt{x} + 5\end{align*}
y=2x√+5 
\begin{align*}y = 3  \sqrt{x}\end{align*}
y=3−x√ 
\begin{align*}y = 4 + 2 \sqrt{x}\end{align*}
y=4+2x√ 
\begin{align*}y = 2 \sqrt{2x + 3} + 1\end{align*}
y=22x+3−−−−−√+1 
\begin{align*}y = 4 + \sqrt{2  x}\end{align*}
y=4+2−x−−−−−√ 
\begin{align*}y = \sqrt{x + 1}  \sqrt{4x  5}\end{align*}
y=x+1−−−−−√−4x−5−−−−−√
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Image Attributions
Here you'll learn what shifts result from performing operations both inside and outside the square root sign of square root functions. You'll also learn how to graph such functions.