How do the values of a, h, and k affect the graph
of the absolute value function
?
Graphing Absolute Value Equations
Parent Function: y = |x|
x
y
-2
2
-1
1
0
0
1
1
2
2
When choosing
your domain,
you want the
expression
inside of the
absolute value
to be 0!
In the same coordinate plane, graph y = a|x| for a = -2, -½,
½, and 2. What effect does a have on the graph of y = |x|? Describe
the domain and range of each function?
Click for graph of
Click for graph of
y = -2|x|
y = -½|x|
x
y
x
y
-2
-4
-2
-1
-1
-2
-1
-1/2
0
0
0
0
1
-2
1
-1/2
2
-4
2
-1
y = ½|x|
y = 2|x|
x
y
x
y
Click for graph of
-2
1
-2
4
Click for graph of
-1
1/2
-1
2
0
0
0
0
1
1/2
1
2
2
1
2
4
How did the coefficients transform the graph of y=|x|?
In the same coordinate plane, graph y = |x – h| for h = -2, 0,
and 2. What effect does h have on the graph of y = |x|?
Describe the domain and range of each function?
y = |x + 2|
x
y
y = |x – 2|
x
-4
0
-3
1
-2
2
-1
3
0
4
Click for Graph
y
Click for Graph
y = |x – 2|
y = |x + 2|
How did the addition or subtraction to/from x inside
the absolute value sign transform the graph of y=|x|?
In the same coordinate plane, graph y = |x| + k for k = -2, 0,
and 2. What effect does k have on the graph of y = |x|?
Describe the domain and range of each function?
y = |x| – 2
y = |x| + 2
x
y
x
y
-2
0
-2
4
-1
-1
-1
3
0
-2
0
2
1
-1
1
3
2
0
2
4
Click for Graph
y = |x| – 2
Click for Graph
y = |x| + 2
How did the addition or subtraction to/from x outside
the absolute value sign transform the graph of y=|x|?
These transformations won't always happen
individually – they can be combined.
Example: y = 2|x – 4| + 3
Notice:
The slope of the
positive part of the
blue equation is 1
and the slope of the
positive part of the
red equation is 2!
List the transformations that were made to the graph of y=|x|.
-vertex moved to the right 4
-vertex moved up 3
-narrower
General Transformation Rules for
Absolute Value Graphs
Parent Function: ALWAYS y = |x|
The graph of y = a|x – h| + k has the following characteristics:
The graph has vertex (h, k) and is symmetric in the line x=h.
The graph is V-shaped. It opens up if a>0 and down if a<0.
The graph is wider than the parent function if |a|<1.
The graph is narrower than the parent function if |a|>1.
click below
open up/down
narrow/wide
(this
represents part
of the slope)
y = a|x - h| + k
click below
left/right
moves up/down
click above
y = 2|x - 4| + 3
vertex: (4,3)
symmetric x=4
opens up 2>0
narrower |2|>1
Example 1 – Graphing an Absolute
Value Function
Graph y = -|x + 2| + 3
x
y
Click For Graph
y = -|x + 2| + 3
a=
h=
k=
Vertex =
Example 2 – Graphing an Absolute
Value Function
Graph y = |x – 2| – 3
x
y
Click For Graph
y = |x – 2| – 3
a=
h=
k=
Vertex =
Example 3 – Writing an Absolute Value
Function
Write the equation of the graph
shown.
Click For Answer!
(2, 1)
(0, -3)
a=
h=
k=
Vertex =
The slope of the positive part is 2, so it
becomes narrower.
It is also shifted 3 units down.
Therefore, the equation is
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