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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

Shifts of Square Root Functions 

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

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

License: CC BY-NC 3.0

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.

 

 

Graphing Multiple Functions 

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

License: CC BY-NC 3.0

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.

Combining Transformations  

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

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:

License: CC BY-NC 3.0

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.

License: CC BY-NC 3.0

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.

 

 

Example

Example 1

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

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:

License: CC BY-NC 3.0

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.

License: CC BY-NC 3.0

Review 

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*}

Review (Answers)

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

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Vocabulary

square root function

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

Transformations

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

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0
  6. [6]^ License: CC BY-NC 3.0

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