What if you had a square root function like \begin{align*}y = \sqrt{2x} + 3\end{align*}. How would you graph that function? After completing this Concept, you'll be able to graph square root functions like this one and compare them to other square root functions.

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CK-12 Foundation: Graphs of Square Root Functions

### Guidance

In this chapter you’ll learn about a different kind of function called the square root function. You’ve seen that taking the square root is very useful in solving quadratic equations. For example, to solve the equation \begin{align*}x^2 = 25\end{align*} we take the square root of both sides: \begin{align*}\sqrt{x^2} = \pm \sqrt{25}\end{align*}, so \begin{align*}x = \pm 5\end{align*}.

A square root function is any function with the form: \begin{align*}y = a \sqrt{f(x)} + c\end{align*} —in other words, any function where an expression in terms of \begin{align*}x\end{align*} is found inside a square root sign (also called a “radical” sign), although other terms may be included as well.

**Graph and Compare Square Root Functions**

When working with square root functions, you’ll have to consider the domain of the function before graphing. The domain is very important because the function is undefined when the expression inside the square root sign is negative, and as a result there will be no graph in whatever region of \begin{align*}x-\end{align*}values makes that true.

To discover how the graphs of square root functions behave, let’s make a table of values and plot the points.

#### Example A

*Graph the function* \begin{align*}y = \sqrt{x}\end{align*}.

**Solution**

Before we make a table of values, we need to find the domain of this square root function. The domain is found by realizing that the function is only defined when the expression inside the square root is greater than or equal to zero. Since the expression inside the square root is just \begin{align*}x\end{align*}, that means the domain is all values of \begin{align*}x\end{align*} such that \begin{align*}x \ge 0.\end{align*}

This means that when we make our table of values, we should pick values of \begin{align*}x\end{align*} that are greater than or equal to zero. It is very useful to include zero itself as the first value in the table and also include many values greater than zero. This will help us in determining what the shape of the curve will be.

\begin{align*}x\end{align*} | \begin{align*}y=\sqrt{x}\end{align*} |
---|---|

0 | \begin{align*}y=\sqrt{0} = 0\end{align*} |

1 | \begin{align*}y=\sqrt{1} = 1\end{align*} |

2 | \begin{align*}y=\sqrt{2} = 1.4\end{align*} |

3 | \begin{align*}y=\sqrt{3} = 1.7\end{align*} |

4 | \begin{align*}y=\sqrt{4} = 2\end{align*} |

5 | \begin{align*}y=\sqrt{5} = 2.2\end{align*} |

6 | \begin{align*}y=\sqrt{6} = 2.4\end{align*} |

7 | \begin{align*}y=\sqrt{7} = 2.6\end{align*} |

8 | \begin{align*}y=\sqrt{8} = 2.8\end{align*} |

9 | \begin{align*}y=\sqrt{9} = 3\end{align*} |

Here is what the graph of this table looks like:

The graphs of square root functions are always curved. The curve above looks like half of a parabola lying on its side, and in fact it is. It’s half of the parabola that you would get if you graphed the expression \begin{align*}y^2 = x\end{align*}. And the graph of \begin{align*}y = - \sqrt{x}\end{align*} is the other half of that parabola:

Notice that if we graph the two separate functions on the same coordinate axes, the combined graph is a parabola lying on its side.

Now let's compare square root functions that are multiples of each other.

#### Example B

*Graph the functions* \begin{align*}y = \sqrt{x}, y=2\sqrt{x}, y=3\sqrt{x},\end{align*} *and* \begin{align*}y = 4\sqrt{x}\end{align*} *on the same graph.*

**Solution**

Here is just the graph without the table of values:

If we multiply the function by a constant bigger than one, the function increases faster the greater the constant is.

#### Example C

*Graph the functions* \begin{align*}y=\sqrt{x}, y= \sqrt{2x}, y=\sqrt{3x},\end{align*} *and* \begin{align*}y=\sqrt{4x}\end{align*} *on the same graph.*

**Solution**

Notice that multiplying the expression *inside* the square root by a constant has the same effect as multiplying by a constant *outside* the square root; the function just increases at a slower rate because the entire function is effectively multiplied by the square root of the constant.

Also note that the graph of \begin{align*}\sqrt{4x}\end{align*} is the same as the graph of \begin{align*}2\sqrt{x}\end{align*}. This makes sense algebraically since \begin{align*}\sqrt{4} = 2\end{align*}.

#### Example D

*Graph the functions* \begin{align*}y=\sqrt{x}, y = \frac{1}{2} \sqrt{x}, y = \frac{1}{3} \sqrt{x},\end{align*} *and* \begin{align*}y = \frac{1}{4} \sqrt{x}\end{align*} *on the same graph.*

**Solution**

If we multiply the function by a constant between 0 and 1, the function increases more slowly the smaller the constant is.

Watch this video for help with the Examples above.

CK-12 Foundation: Graphs of Square Root Functions

### Vocabulary

- A
**square root function**is any function with the form: \begin{align*}y = a \sqrt{f(x)} + c\end{align*} —in other words, any function where an expression in terms of \begin{align*}x\end{align*} is found inside a square root sign (also called a “radical” sign).

### Guided Practice

*Graph the functions*

a) \begin{align*}y=2 \sqrt{x}\end{align*} *and* \begin{align*}y=-2 \sqrt{x}\end{align*} *on the same graph.*

b) \begin{align*}y=\sqrt{x}\end{align*} *and* \begin{align*}y = \sqrt{-x}\end{align*} *on the same graph.*

**Solutions:**

a) If we multiply the whole function by -1, the graph is reflected about the \begin{align*}x-\end{align*}axis.

b)

On the other hand, when just the \begin{align*}x\end{align*} is multiplied by -1, the graph is reflected about the \begin{align*}y-\end{align*}axis. Notice that the function \begin{align*}y=\sqrt{-x}\end{align*} has only negative \begin{align*}x-\end{align*}values in its domain, because when \begin{align*}x\end{align*} is negative, the expression under the radical sign is positive.

### Explore More

Graph the following functions.

- \begin{align*}y = 3\sqrt{x}\end{align*}
- \begin{align*}y = -\frac{1}{2}\sqrt{x}\end{align*}
- \begin{align*}y = \sqrt{4x}\end{align*}
- \begin{align*}y = \sqrt{x}+7\end{align*}
- \begin{align*}y = 2\sqrt{x}-5\end{align*}
- \begin{align*}y = -\sqrt{3x+1}-2\end{align*}

Graph the following functions on the same coordinate axes.

- \begin{align*}y = \sqrt{x}, y = 2.5\sqrt{x}\end{align*} and \begin{align*} y= -2.5\sqrt{x}\end{align*}
- \begin{align*}y = \sqrt{x}, y = 0.3 \sqrt{x}\end{align*} and \begin{align*} y= 0.6\sqrt{x}\end{align*}
- \begin{align*}y = \sqrt{x}, y = \sqrt{x - 5}\end{align*} and \begin{align*} y= \sqrt{x + 5}\end{align*}
- \begin{align*}y = \sqrt{x}, y = \sqrt{x} + 8\end{align*} and \begin{align*} y= \sqrt{x} - 8\end{align*}