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Unit 5 Lesson 2 Graph absolute value functions

Absolute Value Function (parent function): y = |x|

[pic]

Example: Find the vertex of the absolute value function.

1. y = | x – 2| - 4 2. y = | x + 1| + 3

Set x – 2 = 0 and solve. Set x + 1 = 0 and solve

x = 2 x = - 1

Substitute the value of x into the equation. Substitute the value of x into the equation.

y = | 2 – 2| - 4 y = | -1 + 1| + 3

y = -4 y = 3

The vertex is (2, -4). The vertex is (-1, 3).

To graph absolute value functions:

1. Find the vertex.

2. Make a table with the vertex in the middle.

3. Choose x values that are smaller and larger than the x of the vertex.

4. Find your corresponding y values.

5. Graph the table.

Example: Graph y = | x + 2| – 3

|X |Y |

|–5 |0 |

|–4 |–1 |

|–3 |–2 |

|–2 |–3 |

|–1 |–2 |

|0 |–1 |

|1 |0 |

The vertex is (–2, –3). To make the table, three values smaller than –2 will be –3, –4, and –5. Three values larger than –2 will be –1, 0, and 1. Substitute each of the “x” values into the function to find the corresponding “y” values. Once the table is complete, plot each point and draw the graph.

[pic]

The graph of the absolute value parent function can be compressed or stretched by changing the number in front of the absolute value. If the number in front of the absolute value is greater than 1, the graph is vertically stretched. If the number in front of the absolute value is less than one, the graph is vertically compressed.

Ex

|X |Y |

|–2 |3 |

|–1 |1 |

|0 |–1 |

|1 |–3 |

|2 |–1 |

|3 |1 |

|4 |3 |

[pic]

▪ 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.

▪ When you have a function in the form y = a|x| the graph will be stretched if a > 1 or compressed (shrink) is 0 < a < 1.

▪ When you have a function in the form of y = – |x| the graph will reflect (FLIP) over the x-axis.

Absolute Value Functions Practice

Find the vertex of each function.

1. y = | x | + 3 2. y = | x – 5| – 7 3. y = |2x + 1| + 5

Graph each absolute value function. Describe the translation from the parent function for each problem.

4. y = | x | – 1 5. y = | x + 2 | + 1 6. y = – | x – 1 | – 2

7. y = 3 | x | 8. y = 2 | x – 2 | 9. y = [pic]| x + 1 | – 2

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Graphs look like a “v”.

The vertex of an absolute value function is

where the graph is a minimum or a maximum.

This graph is transformed 2 units to the left and 3 units down from the parent function.

This graph is transformed 1 unit to the right, 3 units down and a vertical stretch of 2 from the parent function.

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