In general, the graph of the absolute value function
f (x) = a| x - h| + k
is a shape "V" with vertex (h, k).
To graph the absolute value function, we should be aware of the following terms.
Horizontal Shift :
A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift.
Vertical Shift :
A shift to the input results in a movement of the graph of the function up or down in what is known as a vertical shift.
Stretches and compression :
y = a |x-h| + k
Here a is slope, by observing the rate of change we can fix the value of a.
Reflection :
y = -a |x-h| + k
Reflects it about x-axis
Write the absolute value equation of the following graph.
Example 1 :
Solution :
Vertex of the absolute value function :
(h, k) ==> (0, 0)
From (0, 0) to (1, 2)
Rise is 2 units and run is 1 unit and it is open upward.
Slope (a) = Rise/Run
Slope (a) = 2/1 ==> 2
Applying the the values discussed above, we get
y = a|x - h| + k
y = 2|x - 0| + 0
y = 2|x|
Example 2 :
Solution :
Vertex of the absolute value function :
(h, k) ==> (3, 1)
From (3, 1) to (4, 0)
Rise is 1 unit and run is 1 unit and it is open downward.
Slope (a) = Rise/Run
Slope (a) = 1/1 ==> -1
Applying the the values discussed above, we get
y = a |x-h| + k
y = -1|x-3|+1
So, the required absolute equation for the given graph is
y = -1|x-3|+1
Example 3 :
Solution :
Vertex of the absolute value function :
(h, k) ==> (-2, 0)
From (-2, 0) to (-5, 1)
Rise is 1 unit and run is 2 units and it is open upward.
Slope (a) = Rise/Run
Slope (a) = 1/2 ==> 1/2
Applying the the values discussed above, we get
y = a |x-h| + k
y = (1/2)|x+2|+0
y = (1/2)|x+2|
So, the required absolute equation for the given graph is
y = (1/2)|x+2|
Example 4 :
Solution :
Vertex of the absolute value function :
(h, k) ==> (-1, -1)
From (-1, -1) to (-2, 1)
Rise is 2 units and run is 1 unit and it is open upward.
Slope (a) = Rise/Run
Slope (a) = 2/1 ==> 2
Applying the the values discussed above, we get
y = a |x-h| + k
y = 2|x+1|+(-1)
y = 2|x+1|-1
So, the required absolute equation for the given graph is
y = 2|x+1|-1
Example 5 :
Solution :
Vertex of the absolute value function :
(h, k) ==> (2, 6)
From (2, 6) to (8, 4)
Rise is 1 unit and run is 3 units and it is open down
Slope (a) = Rise/Run
Slope (a) = -1/3
Applying the the values discussed above, we get
y = a |x-h| + k
y = (-1/3)|x-2|+6
y = (1/3)|x-2|+6
So, the required absolute equation for the given graph is
y = (1/3)|x-2|+6
Example 6 :
Solution :
Vertex of the absolute value function :
(h, k) ==> (0, 20)
From (0, 20) to (5, 0)
Rise is 4 units and run is 1 unit and it is opens down.
Slope (a) = Rise/Run
Slope (a) = -4/1 ==> -4
Applying the the values discussed above, we get
y = a|x - h| + k
y = -4|x -0| + 20
y = -4|x| + 20
So, the required absolute equation for the given graph is
y = -4|x| + 20
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