Notes for Lesson 5-8: Graphing Absolute Value Functions

5-8.1 – The parent function of an absolute value function

Vocabulary: y Translation – A shift in a graph vertically, horizontally or both 10

The base or parents function of an absolute value function if which 5 looks like the graph to the right. Every absolute value graph will look like 0 this one to some extent. x

However, when we do things to the basic function that changes the graph. -5 These changes are called transformations and one of the most common is a -10 translation. This moves the graph up, down, left or right. These changes can -10 -5 0 5 10 be recognized by changes made in the equation.

5-8.2 – Graphing a vertical translation

If there is a value being added onto the end of the absolute value function then graph will be moved up or down the number of spaces of that value. We say this value is k and the equation would look like this: . So to graph this, we can graph our basic function only moved over k number of spaces.

Example:

y y 10 10

0 0 x x

-10 -10 -5 0 5 10 -10 -10 -5 0 5 10 5-8.3 – Graphing a horizontal translation

If there is a value being added to the variable inside the absolute value bars, this will move the graph horizontally, However, it moves the opposite direction of the sign of the number. So if you add a number the graph will move left and if you add a negative (or subtract a number) then it will move right.

Examples: y y 10 10

0 0 x x

-10 -10 -10 -5 0 5 10 -10 -5 0 5 10 5-8.4 – Graphing an absolute value with both translations You can have graphs that move in multiple directions.

Examples:

The -1 moves it right 1 space The +2 moves it left 2 spaces The +2 moves it up 2 spaces The – 1 moves it down 1 space The – outside also flips the V downward

y y 10 10

0 0 x x

-10 -10 -5 0 5 10 -10 -10 -5 0 5 10

5-8.5 – Step Functions

Vocabulary: Piecewise Function – A function with different rules for different parts of its domain Step Function – A function that pairs every number in an interval with a single value.

An absolute value function is an example of a piece wise function. This is because for all numbers the absolute value is simply but for all values less than 0 the absolute value is which changes it to positive.

Another piecewise function is called a step function, which is where certain intervals of numbers have a common output.

Example:

A school will charter buses so that the students can attend a field trip. Each bus can hold up to 60 students. Make a graph that shows the number of buses needed for x number of students.

You need 0 busses for 0 students, but for 1 to 60 students you need 1bus, for 61 to 120you need 2 busses, 121 to 180 you need 3 busses And so forth.