Which of These is the Absolute Value Parent Function

Absolute Value Functions

An absolute value function is a role that contains an algebraic expression within absolute value symbols. Recall that the accented value of a number is its distance from

$\mathrm{}$

on the number line.

The absolute value parent role, written as

$f$

(
10
)

=

|

x

|

, is defined as

$$
f

(
10
)

=

{



x
      if
x
>






      if
10
=





−
x
   if
ten
<

To graph an absolute value role, choose several values of

$x$

and find some ordered pairs.

$x$	$y$ = | 10 |
−2	2
−ane	1
0	0
one	1
2	2

Plot the points on a coordinate plane and connect them.

Observe that the graph is V-shaped.

(

$1$

) The vertex of the graph is

$$
(


,


)

.

(

$\mathrm{two}$

) The centrality of symmetry (

$x$
=

or

$y$

-axis) is the line that divides the graph into two congruent halves.

(

$3$

) The domain is the prepare of all real numbers.

(

$4$

) The range is the set of all real numbers greater than or equal to

$\mathrm{}$

. That is,

$y$
≥

.

(

$\mathrm{five}$

) The

$x$

-intercept and the

$y$

-intercept are both

$\mathrm{}$

.

Vertical Shift

To interpret the accented value part

$f$

(
x
)

=

|

ten

|

vertically, you tin can utilize the function

$m$

(
x
)

=
f

(
10
)

+
k

.

When

$k$
>

, the graph of

$\mathrm{grand}$

(
ten
)

translated

$k$

units up.

When

$\mathrm{yard}$
<

, the graph of

$g$

(
x
)

translated

$\mathrm{one\; thousand}$

units down.

Horizontal Shift

To interpret the absolute value function

$f$

(
x
)

=

|

10

|

horizontally, yous can employ the function

$\mathrm{yard}$

(
10
)

=
f

(

10
−
h

)

.

When

$h$
>

, the graph of

$f$

(
x
)

is translated

$h$

units to the right to get

$g$

(
x
)

.

When

$h$
<

, the graph of

$f$

(
ten
)

is translated

$h$

units to the left to get

$g$

(
10
)

.

Stretch and Compression

The stretching or compressing of the absolute value function

$y$
=

|

x

|

 is defined by the part

$y$
=
a

|

x

|

 where

$a$

is a constant. The graph opens up if

$a$
>

 and opens down when

$a$
<

.

For accented value equations multiplied by a constant

$($
for case,
y
=
a

|


x


|

)

,if

$\mathrm{}$
<
a
<
ane

, then the graph is compressed, and if

$a$
>
1

, it is stretched. As well, if a is negative, and then the graph opens downward, instead of upwards as usual.

More mostly, the class of the equation for an absolute value office is

$y$
=
a

|


x
−
h


|

+
k

. Also:

• The vertex of the graph is
  $$
  (
  
  h
  ,
  chiliad
  
  )
  
  

  .

• The domain of the graph is fix of all real numbers and the range is
  $y$
  ≥
  k
  

   when

  $a$
  >
  
  

  .

• The domain of the graph is set of all real numbers and the range is
  $y$
  ≤
  k
  

   when

  $a$
  <
  
  

  .

• The axis of symmetry is
  $x$
  =
  h
  

  .

• It opens up if
  $a$
  >
  
  

   and opens down if

  $a$
  <
  
  

  .

• The graph
  $y$
  =
  
  |
  
  x
  
  |
  
  

  can be translated

  $h$
  

  units horizontally and

  $\mathrm{thou}$
  

  units vertically to get the graph of

  $y$
  =
  a
  
  |
  
  
  x
  −
  h
  
  
  |
  
  +
  k
  

  .

• The graph
  $y$
  =
  a
  
  |
  
  ten
  
  |
  
  

   is wider than the graph of

  $y$
  =
  
  |
  
  x
  
  |
  
  

  if

  $$
  |
  
  a
  
  |
  
  <
  1
  

   and narrower if

  $$
  |
  a
  |
  
  >
  ane
  

  .