Concept Byte: Characteristics of Absolute Value Graphs

Use With Lesson 5-8

EXTENSION

In previous lessons you explored characteristics of linear graphs. Here you will explore characteristics of absolute value graphs of the form y = a | x − h | + k . y equals eh vertical line x minus h vertical line plus k .

For a linear graph you can identify the x- and y-intercepts, the domain and range, and the slope. For an absolute value graph, you can also identify the direction the graph opens and the vertex. The vertex of an absolute value graph is the point at which the graph changes direction. The graph of y = a | x − h | + k y equals eh vertical line x minus h vertical line plus k  has vertex (h, k).

The graph of an absolute value function will always have one y-intercept, but it can have zero, one, or two x-intercepts. An absolute value graph has a different slope for each branch. The branches are the two rays on either side of the vertex.

Example

Graph y = | x + 1 | − 2 . bold italic y equals vertical line bold italic x plus 1 vertical line negative 2 .  What are the slope of each branch, the x- and y-intercepts, the vertex, and the domain and range?

Image Long Description

• Step 1 Plot the vertex ( − 1 , − 2 ) . open negative 1 comma negative 2 close .
• Step 2 Use the equation to find a point on either side of the vertex.
• Step 3 Draw the two branches of the graph.
• The domain is all real numbers. The range is all real numbers greater than or equal to − 2 . negative 2 .

Exercises

1. 1. Graph y = − | x − 3 | − 4 , y = | x − 3 | − 4 , y = − 2 | x + 3 | − 4 , y equals negative vertical line x minus 3 vertical line negative 4 comma y equals vertical line x minus 3 vertical line negative 4 comma y equals negative 2 vertical line x plus 3 vertical line negative 4 comma  and y = 2 | x + 3 | − 4 . y equals 2 vertical line x plus 3 vertical line negative 4 .
   2. Which graphs open up and which graphs open down?
   3. Reasoning How does the sign of a affect the direction the graph opens?
   4. What are the slopes of the left and right branches of each graph?
   5. Reasoning How does the slope of the left branch relate to the slope of the right branch? How does a relate to the slope of the branches?
2. 1. Graph y = − 2 | x − 1 | + 4 . y equals negative 2 vertical line x minus 1 vertical line plus 4 .
   2. What is the vertex of the graph?
   3. What are the domain and range of the function?
   4. What are the x- and y-intercepts?
   5. Reasoning How can you use the vertex and the sign of a to determine the range of the absolute value function?

Table of Contents