### Graphs of Square Root Functions

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

A square root function is any function with the form:

**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

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

#### Graphing Functions

Graph the function

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

This means that when we make our table of values, we should pick values of

0 | |

1 | |

2 | |

3 | |

4 | |

5 | |

6 | |

7 | |

8 | |

9 |

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

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.

#### Graphing Multiple Functions on the Same Graph

1. Graph the functions

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.

2. Graph the functions

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

3. Graph the functions

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

### Examples

Graph the functions

**Example 1**

If we multiply the whole function by -1, the graph is reflected about the

#### Example 2

On the other hand, when just the

### Review

Graph the following functions.

y=3x√ y=−12x√ y=4x−−√ y=x√+7 y=2x√−5 y=−3x+1−−−−−√−2

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*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 11.1.