GRAPHING ABSOLUTE VALUE FUNCTIONS

The following steps will be useful to graph absolute value functions.

Step 1 :

Before graphing any absolute value function, first we have to graph the parent function :

y = |x|

Its vertex is (0,0)

Let us take some random values for x.

x = -3 ----> y = |-3| = 3 ----> (-3, 3)

x = -2 ----> y = |-2| = 2 ----> (-2, 2)

x = -1 ----> y = |-1| = 1 ----> (-1, 1)

x = 0 ----> y = |0| = 0 ----> (0, 0)

x = 1 ----> y = |1| = 1 ----> (1, 1)

x = 2 ----> y = |2| = 2 ----> (2, 2)

x = 3 ----> y = |3| = 3 ----> (3, 3)

If we plot these points on the graph sheet, we will get a graph as given below.

When we look at the above graph, clearly the vertex is

(0, 0)

Step 2 :

Write the given absolute value function as

y = |x - h| + k

Vertex is (h, k).

According to the vertex, we have to shift the above graph.

Note :

If we have negative sign in front of absolute sign, we have to flip the curve over.

Example :

y = -|x|

In each of the following examples, graph the given absolute value function.

Example 1 :

y = |x - 1|

Solution :

y = |x - 1| ----> y = |x - 1| + 0

Compare :

y = |x - h| + k

y = |x - 1| + 0

Vertex (h, k) = (1, 0).

Example 2 :

y = |x - 1| - 2

Solution :

Compare :

y = |x - h| + k

y = |x - 1| - 2

Vertex (h, k) = (1, -2).

Example 3 : 

y = |x + 3| + 3

Solution :

Compare :

y = |x - h| + k

y = |x + 3| + 3

Vertex (h, k) = (-3, 3).

Example 4 :

y = |x - 2|

Solution :

y = |x - 2| ----> y = |x - 2| + 0

Compare :

y = |x - h| + k

y = |x - 2| + 0

Vertex (h, k) = (2, 0).

Example 5 :

y = |x + 4| + 3

Solution :

Compare :

y = |x - h| + k

y = |x + 4| + 3

Vertex (h, k) = (-4, 3).

Example 6 :

y = |x - 4| - 4

Solution :

Compare :

y = |x - h| + k

y = |x - 4| - 4

Vertex (h, k) = (4, -4).

Example 7 :

y = -|x - 2| - 2

Solution :

Compare :

y = |x - h| + k

y = -|x - 2| - 2

Vertex (h, k) = (2, -2).

Because there is negative sign in front of the absolute sign, we have to flip the curve over.  

Example 8 :

y = -|x - 4|

Solution :

y = -|x - 4| ----> y = -|x - 4| + 0

Compare :

y = |x - h| + k

y = -|x - 4| + 0

Vertex (h, k) = (4, 0).

Because there is negative sign in front of the absolute sign, we have to flip the curve over. 

Example 9 :

y = -|x| + 2

Solution :

y = -|x| + 2 ----> y = -|x - 0| + 2

Compare :

y = |x - h| + k

y = -|x - 0| + 2

Vertex (h, k) = (0, 2).

Because there is negative sign in front of the absolute sign, we have to flip the curve over.  

Example 10 :

y = -|x + 1| + 3

Solution :

Compare :

y = |x - h| + k

y = -|x + 1| + 3

Vertex (h, k) = (-1, 3).

Because there is negative sign in front of the absolute sign, we have to flip the curve over. 

Example 11 :

y = -|x| + 4

Solution :

y = -|x| + 4 ----> y = -|x - 0| + 4

Compare :

y = |x - h| + k

y = -|x - 0| + 4

Vertex (h, k) = (0, 4).

Because there is negative sign in front of the absolute sign, we have to flip the curve over. 

Example 12 :

y = -|x + 1| - 1

Solution :

Compare :

y = |x - h| + k

y = -|x + 1| - 1

Vertex (h, k) = (-1, -1).

Because there is negative sign in front of the absolute sign, we have to flip the curve over.

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