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

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 #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 ,

or, alternatively written,

The graph of  is the same as that of , after it shifts 10 units left (  ), it flips vertically (negative symbol), and it shifts up 10 units (the second ). The flip does not affect the position of the vertex, but the shifts do; the vertex of the graph of  is at .

Example Question #7 : Graphing Absolute Value Functions

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