To see what the

graph of y = |x| looks like, let’s create a

table of values.

To

graph these values, simply plot the points and see what happens.

Whenever you have an

absolute value graph, the general shape will look like a “v” (or in some cases, an upside down “v” as we will see later).

**Let's Practice:**- Graph y = |x+2|

We know what the general shape should look like, but let’s create a table of values to see exactly how this graph will look.

**x** | **y** |

-3 | |-3 + 2| = |-1| = 1 |

-2 | |-2 + 2| = |0| = 0 |

-1 | |-1 + 2| = |1| = 1 |

0 | |0 + 2| = |2| = 2 |

1 | |1 + 2| = |3| = 3 |

2 | |2 + 2| = |4| = 4 |

3 | |3 + 2| = |5| = 5 |

So our graph of y = |x + 2| looks like

Notice that the graph in this example looks almost identical to the graph of y = |x| except that it was shifted to the left 2 units. This will be important as we try to make generalizations later in the lesson.

- Graph y = |x| - 4

The table of values looks like this:

**x** | **y** |

-5 | 5 - 4 = 1 |

-4 | 4 - 4 = 0 |

-3 | 3 - 4 = -1 |

-2 | 2 - 4 = -2 |

-1 | 1 - 4 = -3 |

0 | 0 - 4 = -4 |

1 | 1 - 4 = -3 |

Which makes the graph look like this:

Notice that the graph in this example is the same shape as except that it has been moved down 4 units.

- Graph y = -|x|

In creating the table of values, be careful of your order of operations. You should find the absolute value of x first and then change the sign of that answer.

**x** | **|x|** | **y** |

-3 | 3 | -3 |

-2 | 2 | -2 |

-1 | 1 | -1 |

0 | 0 | 0 |

1 | 1 | -1 |

2 | 2 | -2 |

3 | 3 | -3 |

So the graph of looks like:

In this example, we have the exact same shape as the graph of y = |x| only the “v” shape is upside down now.

Based on the examples we’ve seen so far, there appears to be a

pattern when it comes to graphing

absolute value functions.

- When you have a function in the form
**y = |x + h|** the graph will move h units to the left.

When you have a function in the form **y = |x - h| **the graph will move h units to the right.

- When you have a function in the form
**y = |x| + k** the graph will move up k units.

When you have a function in the form **y = |x| - k** the graph will move down k units.

- If you have a negative sign in front of the absolute value, the graph will be reflected, or flipped, over the x-axis.

Keep in mind that you can also have combinations that change the

absolute value graph more than once. You can practice these transformations with this

EXCEL Modeling worksheet.