Algebra 1 Common Core Student Edition, Grade 8-9

. Co m m o n Co r e

Concept Byte | Characteristics of

.. , r n m I » I i piecewise-def ined f unct ions, indudinq step
Use W ith Lesson 5 8 A functions and absolute value functions.
Absolute Value Graphs MP 7

In previous lessons you explored characteristics of linear graphs. H ere you will explore

characteristics of absolute value graphs of th e form y = a \x - h\ + k.

For a linear graph you can identify th e x- an d y-intercepts, th e d o m ain a n d range, an d

the slope. For an absolute value graph, you can also identify th e direction th e graph

opens and the vertex. The vertex of an absolute value g raph is th e p o in t at w hich the

graph changes direction. The graph of y = a \x — h\ + k has vertex {h, k).

The graph of an absolute value function will always have o n e y-intercept, b u t it can

have zero, one, or two x-intercepts. An absolute value graph has a different slope for

each branch. The b ran ch es are th e two rays o n eith er side of th e vertex.

Ex a m p l e

Graph y = \ x + 11 - 2. What are the slope of each branch,

the vertex, and the domain and range?

Step 1 Plot the vertex (-1 , -2 ).

S te p 2 Use th e equ atio n to find a p o in t on eith er side of

the vertex.

Step 3 Draw th e two b ra n ch es of th e graph.

The dom ain is all real num bers. The range is all real

num bers greater th a n or equal to - 2.

A.

Ex e r c i se s

1. a. G raph y = — \x — 3| - 4, y= \x — 3| - 4, y = —2|x + 3| - 4, a n d
   y = 2 |x + 3| - 4.
   b. W hich graphs o p en up a n d w hich graphs o p en down?
   c. R easo ning How does th e sign of a affect th e direction th e graph opens?
   d. W hat are th e slopes of the left an d right b ra n ch es of each graph?
   e. Reasoning How does th e slope of th e left b ra n c h relate to th e slope of th e right
   branch? How does a relate to th e slope of th e branches?


2. a. Graph y = -2 |x- 11 + 4.
   b. W hat is th e vertex of th e graph?
   c. W hat are the do m ain an d range of th e function?
   d. W hat are th e x- and y-intercepts?
   e. Reasoning How can you u se th e vertex an d th e sign of a to d e t e r m in e th e
   range of th e absolute value function?

Concept Byte Characteristics of Absolute Value Graphs j 351

the x- and y-intercepts,

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slope: -1

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