Key Concepts

Square root.

Square root functions are closely related to quadratic functions, because as you may have already noticed, square roots "undo" squaring a number, and vice versa, squaring a number "undoes" a square root. As a result, the square root family of functions have graphs that somewhat resemble the quadratic graphs with two notable exceptions -- 1) they're sideways and 2) it's only half the graph.

The "parent" functions for the square root family is

\(f(x) = \sqrt{x}.\)

Notice how the graph looks like half of the quadratic graph that's been turned on its side. The basic shape is a "swoosh", but just like with the previous families, other functions in this family may be modified so that their graph is flipped over vertically or horizontally, moved around, or stretched.

There are three really important things about the square root functions to point out.

1. Remember that when we take the square root of a number, we get both a positive and a negative answer. However, when we are thinking about square root as a function, we only want it to have ONE output, not two. In order for a square root to be considered a function, we only consider the positive results from square roots.
2. Since every squared real number produces a positive number, going the opposite direction with a square root means we cannot take the square root of a negative number and get a real number back. That is, when we are looking at square roots as a function, we only take the square roots of positive numbers or 0.
3. Always make the first point you graph the place where you are taking the square root of 0 to make sure the "swoosh" starts in the right place.

When graphing these types of functions, it's pretty common to wind up with decimals for outputs because the inputs may not always result in taking "nice" square roots. If we want, we can choose "convenient" numbers to plug in that will always cause us to take square roots of these "nice" numbers (e.g., 0, 1, 4, 9, 16, etc), or we can just deal with the decimals. It's entirely up to you!

Our first example has been moved horizontally.

Example \(\PageIndex{12}\)

Graph the following square root function using a table: \(f(x)=\sqrt{x + 1}\)

Solution

We first make the table with points to plot. Notice that since inside the square root is \(x + 1\), starting at \(x = -1\) is fine because \(-1 + 1 = 0\), which we are allowed to take the square root of! This will be the first point we want to include on the table. From here, we'd like to take square roots of the "nice" squared numbers 1, 4, and 9. So we want to plug in \(x = 0, 3,\) and 8.

\(\begin{array} {|c|c|}\hline x & f(x) & (x, f(x)) \\ \hline -1 & \sqrt{-1 + 1} = \sqrt{0} = 0 & (-1, 0) \\ \hline 0 & \sqrt{0 + 1} = \sqrt{1} = 1 & (0, 1) \\ \hline 3 & \sqrt{3 + 1} = \sqrt{4} = 2 & (3, 2) \\ \hline 8 & \sqrt{8 + 1} = \sqrt{9} = 3 & (8, 3) \\ \hline \end{array}\)

Now we plot the points.

We complete the graph by knowing that it starts at (-1, 0), and then must follow the "swoosh" shape to the right.

Let's look at a version of the graph that has been moved around both horizontally and vertically.

Example \(\PageIndex{13}\)

Graph \(f(x)=\sqrt{x+1}-2\) by plotting points.

Answer

In regards to the table, notice that once again since inside the square root is \(x + 1\), starting at \(x = -1\) is fine because \(-1 + 1 = 0\), which we are allowed to take the square root of! This will be the first point we want to include on the table. From here, we'd like to take square roots of the "nice" squared numbers 1, 4, and 9. So we want to plug in \(x = 0, 3,\) and 8.

Table	Graph
\(\begin{array} {|c|c|}\hline x & f(x) & (x, f(x)) \\ \hline -1 & \sqrt{-1 + 1} - 2 = \sqrt{0} - 2 = -2 & (-1, -2) \\ \hline 0 & \sqrt{0 + 1} - 2= \sqrt{1} - 2 = -1 & (0, -1) \\ \hline 3 & \sqrt{3 + 1} - 2= \sqrt{4} - 2= 0& (3, 0) \\ \hline 8 & \sqrt{8 + 1} - 2= \sqrt{9} - 2= 1 & (8, 1) \\ \hline \end{array}\)

Finally, let's look at an example where the square root "swoosh" has been moved horizontally, vertically, and flipped over.

Exercise \(\PageIndex{14}\)

Graph the following square root function using a table: \(f(x) = -\sqrt{x+1}-2\)

Answer

Table	Graph
\(\begin{array} {|c|c|}\hline x & f(x) & (x, f(x)) \\ \hline -1 & -\sqrt{-1 + 1} - 2 = -\sqrt{0} - 2 = -2 & (-1, -2) \\ \hline 0 & -\sqrt{0 + 1} - 2= -\sqrt{1} - 2 = -3 & (0, -3) \\ \hline 3 & -\sqrt{3 + 1} - 2= -\sqrt{4} - 2= -4& (3, -4) \\ \hline 8 & -\sqrt{8 + 1} - 2= -\sqrt{9} - 2= -5 & (8, -5) \\ \hline \end{array}\)