# 7.4: Graphing Square Root Functions

**At Grade**Created by: CK-12

**Practice**Graphs of Square Root Functions

Mrs. Garcia has assigned her student the function

Alendro says that because it is a square root function, it can only have positive values and therefore his graph is only in the first quadrant.

Dako says that because of the two negative sign, all *y* values will be positive and therefore his graph is in the first and second quadrants.

Marisha says they are both wrong. Because it is a negative square root function, her graph is in the third and fourth quadrants.

Which one of them is correct?

### Graphing Square Root Functions

A square root function has the form

x |
y |
---|---|

16 | 4 |

9 | 3 |

4 | 2 |

1 | 1 |

0 | 0 |

-1 | und |

Notice that this shape is half of a parabola, lying on its side. For

Let's graph

To graph this function, draw a table.

x |
y |
---|---|

2 | 5 |

3 | 6 |

6 | 7 |

11 | 8 |

After plotting the points, we see that the shape is exactly the same as the parent graph. It is just shifted up 5 and to the right 2. Therefore, we can conclude that **horizontal shift** and **vertical shift**.

The domain is all real numbers such that

Now's let's graph

From the previous problem, we already know that there is going to be a horizontal shift to the left one unit. The 3 in front of the radical changes the width of the function. Let’s make a table.

x |
y |
---|---|

0 | |

0 | 3 |

3 | 6 |

8 | 9 |

15 | 12 |

Notice that this graph grows much faster than the parent graph. Extracting

Finally, let's graph

Extracting

x |
y |
---|---|

2 | 3 |

3 | 2 |

6 | 1 |

11 | 0 |

18 | -1 |

The negative sign in front of the radical, we now see, results in a reflection over

**Using the graphing calculator:** If you wanted to graph this function using the TI-83 or 84, press **2nd** **GRAPH** and adjust the window.

### Examples

#### Example 1

Earlier, you were asked to determine which student was correct.

If you graph the function

#### Example 2

Evaluate

Plug in

**Graph the following square root functions. Describe the relationship to the parent graph and find the domain and range. Use a graphing calculator for Example 5.**

#### Example 3

Graph

Here, the negative is under the radical. This graph is a reflection of the parent graph over the

The domain is all real numbers less than or equal to zero. The range is all real numbers greater than or equal to zero.

#### Example 4

Graph

The starting point of this function is

The domain is all real numbers greater than or equal to -3. The range is all real numbers greater than or equal to zero.

#### Example 5

Graph

Using the graphing calculator, the function should be typed in as:

### Review

Evaluate the function, *x*.

f(3) f(6) f(13) - What is the domain of this function?

Graph the following square root functions and find the domain and range. Use your calculator to check your answers.

f(x)=x+2−−−−−√ y=x−5−−−−−√−2 - \begin{align*}y=-2 \sqrt{x+1}\end{align*}
- \begin{align*}f(x)=1+ \sqrt{x-3}\end{align*}
- \begin{align*}f(x)=\frac{1}{2} \sqrt{x+8}\end{align*}
- \begin{align*}f(x)=3 \sqrt{x+6}\end{align*}
- \begin{align*}y=2 \sqrt{1-x}\end{align*}
- \begin{align*}y=\sqrt{x+3}-5\end{align*}
- \begin{align*}f(x)=4 \sqrt{x+9}-8\end{align*}
- \begin{align*}y=- \frac{3}{2} \sqrt{x-3}+6\end{align*}
- \begin{align*}y=-3 \sqrt{5-x}+7\end{align*}
- \begin{align*}f(x)=2 \sqrt{3-x}-9\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 7.4.

### Notes/Highlights Having trouble? Report an issue.

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General Equation for a Square Root Function

The general equation for a square root function is where is the horizontal shift and is the vertical shift.square root function

A square root function is a function with the parent function .starting point

The starting point is the initial point of a square root function, .### Image Attributions

Here you'll learn how to graph a square root function with and without a calculator.

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