Mrs. Garcia has assigned her student the function \begin{align*}y = -\sqrt{x + 2} - 3\end{align*}

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?

### Guidance

A square root function has the form \begin{align*}y=a \sqrt{x-h}+k\end{align*}

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

#### Example A

Graph \begin{align*}y= \sqrt{x-2}+5\end{align*}

**Solution:** To graph this function, draw a table. \begin{align*}x=2\end{align*}

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 \begin{align*}h\end{align*}**horizontal shift** and \begin{align*}k\end{align*}**vertical shift**.

The domain is all real numbers such that \begin{align*}x \ge 2\end{align*}

#### Example B

Graph \begin{align*}y=3 \sqrt{x+1}\end{align*}

**Solution:** From the previous example, 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 |
---|---|

\begin{align*}-1\end{align*} |
0 |

0 | 3 |

3 | 6 |

8 | 9 |

15 | 12 |

Notice that this graph grows much faster than the parent graph. Extracting \begin{align*}(h, k)\end{align*}

#### Example C

Graph \begin{align*}f(x)=- \sqrt{x-2}+3\end{align*}

**Solution:** Extracting \begin{align*}(h, k)\end{align*}

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 \begin{align*}x\end{align*}

** Using the graphing calculator:** If you wanted to graph this function using the TI-83 or 84, press \begin{align*}Y=\end{align*}

**2nd**\begin{align*}x^2\end{align*}

**GRAPH**and adjust the window.

**Intro Problem Revisit** If you graph the function \begin{align*}y = -\sqrt{x + 2} - 3\end{align*}

### Guided Practice

1. Evaluate \begin{align*}y=-2 \sqrt{x-5}+8\end{align*}

Graph the following square root functions. Describe the relationship to the parent graph and find the domain and range. Use a graphing calculator for #3.

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

3. \begin{align*}f(x)= \frac{1}{2} \sqrt{x+3}\end{align*}

4. \begin{align*}f(x)=-4 \sqrt{x-5}+1\end{align*}

#### Answers

1. Plug in \begin{align*}x=9\end{align*}

\begin{align*}y=-2 \sqrt{9-5}+8=-2 \sqrt{4}+8=-2(2)+8=-4+8=-4\end{align*}

2. Here, the negative is under the radical. This graph is a reflection of the parent graph over the \begin{align*}y\end{align*}

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

3. The starting point of this function is \begin{align*}(-3, 0)\end{align*}

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

4. Using the graphing calculator, the function should be typed in as: \begin{align*}Y=-4 \sqrt{\;\;}(X-5) + 1\end{align*}

### Explore More

Evaluate the function, \begin{align*}f(x)=-\sqrt{x-4}+3\end{align*} for the following values of *x*.

- \begin{align*}f(3)\end{align*}
- \begin{align*}f(6)\end{align*}
- \begin{align*}f(13)\end{align*}
- 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.

- \begin{align*}f(x)=\sqrt{x+2}\end{align*}
- \begin{align*}y=\sqrt{x-5}-2\end{align*}
- \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 Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.4.