# Algebra II : Graphing Absolute Value Functions

## Example Questions

### Example Question #1 : Graphing Absolute Value Functions

Refer to the above figure.

Which of the following functions is graphed?

Explanation:

Below is the graph of :

The given graph is the graph of  reflected in the -axis, then translated up 6 units. This graph is

, where .

The function graphed is therefore

### Example Question #2 : Graphing Absolute Value Functions

Refer to the above figure.

Which of the following functions is graphed?

Explanation:

Below is the graph of :

The given graph is the graph of  reflected in the -axis, then translated left 2 units (or, equivalently, right  units. This graph is

, where .

The function graphed is therefore

### Example Question #3 : Graphing Absolute Value Functions

Refer to the above figure.

Which of the following functions is graphed?

The correct answer is not given among the other responses.

Explanation:

Below is the graph of :

The given graph is the graph of   translated by moving the graph 7 units left (that is,  unit right) and 2 units down (that is,  units up)

The function graphed is therefore

where . That is,

### Example Question #4 : Graphing Absolute Value Functions

What is the equation of the above function?

Explanation:

The formula of an absolute value function is  where m is the slope, a is the horizontal shift and b is the vertical shift. The slope can be found with any two adjacent integer points, e.g.  and , and plugging them into the slope formula, , yielding . The vertical and horizontal shifts are determined by where the crux of the absolute value function is. In this case, at , and those are your a and b, respectively.

### Example Question #5 : Graphing Absolute Value Functions

Which of the following absolute value functions is represented by the above graph?

The equation cannot be determined from the graph.

Explanation:

The equation can be determined from the graph by following the rules of transformations; the base equation is:

The graph of this base equation is:

When we compare our graph to the base equation graph, we see that it has been shifted right 3 units, up 1 unit, and our graph has been stretched vertically by a factor of 2. Following the rules of transformations, the equation for our graph is written as:

### Example Question #6 : Graphing Absolute Value Functions

Give the vertex of the graph of the function .

None of the other choices gives the correct response.

Explanation:

Let

The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates . In terms of ,

The graph of this function can be formed by shifting the graph of  left 6 units ( ) and down 7 units (). The vertex is therefore located at .

### Example Question #7 : Graphing Absolute Value Functions

Give the vertex of the graph of the function .

None of the other choices gives the correct response.