# GRAPHING ABSOLUTE VALUE FUNCTIONS

Graphing Absolute Value Functions :

In this section, we will learn, how to graph, absolute value functions.

## Graphing Absolute Value Functions - Steps

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 -----> (-3, 3)

x  =  - 1 ----->  y  =  |-1|  =  1 -----> (-3, 3)

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 - k  =  |x - h|

Step 3 :

To get the vertex of the absolute value function above, equate (x - h) and (y - k) to zero,

That is,

x - h  =  0  and  y - k  =  0

x  =  h  and  y  =  k

Therefore, the 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|

## Graphing Absolute Value Functions - Examples

Example 1 :

Graph the absolute value function given below.

y  =  |x - 1|

Solution :

The given absolute value function is in the form :

y - k  =  |x - h|

That is,

y  =  |x - 1|

To get the vertex, equate (x - 1) and y to zero.

x - 1  =  0  and  y  =  0

x  =  1  and  y  =  0

Therefore,  the vertex is

(1, 0)

So, the graph of the given absolute value function is Example 2 :

Graph the absolute value function given below.

y  =  |x - 1| - 2

Solution :

Write the given absolute value function in the form :

y - h  =  |x - h|

That is,

y  =  |x - 1| - 2

y + 2  =  |x - 1|

To get the vertex, equate (x - 1) and (y + 2) to zero.

x - 1  =  0  and  y + 2  =  0

x  =  1  and  y  =  -2

Therefore, the vertex is

(1, -2)

So, the graph of the given absolute value function is Example 3 :

Graph the absolute value function given below.

y  =  |x + 3| + 3

Solution :

Write the given absolute value function in the form :

y - h  =  |x - h|

That is,

y  =  |x + 3| + 3

Subtract 3 from each side.

y - 3  =  |x + 3|

To get the vertex, equate (x + 3) and (y - 3) to zero.

x + 3  =  0  and  y - 3  =  0

x  =  -3  and  y  =  3

Therefore, the vertex is

(-3, 3)

So, the graph of the given absolute value function is Example 4 :

Graph the absolute value function given below.

y  =  |x - 2|

Solution :

The given absolute value function is in the form :

y - k  =  |x - h|

That is,

y  =  |x - 2|

To get the vertex, equate (x - 2) and y to zero.

x - 2  =  0  and  y  =  0

x  =  2  and  y  =  0

Therefore, the vertex is

(2, 0)

So, the graph of the given absolute value function is Example 5 :

Graph the absolute value function given below.

y  =  |x + 4| + 3

Solution :

Write the given absolute value function in the form :

y - h  =  |x - h|

That is,

y  =  |x + 4| + 3

Subtract 3 from each side.

y - 3  =  |x + 4|

To get the vertex, equate (x + 4) and (y - 3) to zero.

x + 4  =  0  and  y - 3  =  0

x  =  -4  and  y  =  3

Therefore, the vertex is

(-4, 3)

So, the graph of the given absolute value function is Example 6 :

Graph the absolute value function given below.

y  =  |x - 4| - 4

Solution :

Write the given absolute value function in the form :

y - h  =  |x -h|

That is,

y  =  |x - 4| - 4

y + 4  =  |x - 4|

To get the vertex, equate (x - 4) and (y + 4) to zero.

x - 4  =  0  and  y + 4  =  0

x  =  4  and y  =  -4

Therefore, the vertex is

(4, -4)

So, the graph of the given absolute value function is Example 7 :

Graph the absolute value function given below.

y  =  -|x - 2| - 2

Solution :

Write the given absolute value function in the form

y - h  =  |x - h|

That is,

y  =  -|x - 2| - 2

y + 2  =  -|x - 2|

To get the vertex, equate (x - 2) and (y + 2) to zero.

x - 2  =  0  and  y + 2  =  0

x  =  2  and  y  =  -2

Therefore, the vertex is

(2, -2)

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

So, the graph of the given absolute value function is Example 8 :

Graph the absolute value function given below.

y  =  -|x - 4|

Solution :

The given function is in the form :

y - k  =  |x - h|

That is,

y  =  -|x - 4|

To get the vertex, equate (x - 4) and y to zero.

x - 4  = 0  and  y  =  0

x  =  4  and  y  =  0

Therefore, the vertex is

(4, 0)

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

So, the graph of the given absolute value function is Example 9 :

Graph the absolute value function given below.

y  =  -|x| + 2

Solution :

Write the given absolute value function in the form :

y - h  =  |x - h|

That is,

y  =  -|x| + 2

Subtract 2 from each side.

y - 2  =  -|x|

To get the vertex, equate x and (y - 2) to zero.

x  = 0  and  y - 2  =  0

x  =  0  and  y  =  2

Therefore, the vertex is

(0, 2)

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

Hence, the graph of the given absolute value function is Example 10 :

Graph the absolute value function given below.

y  =  -|x + 1| + 3

Solution :

Write the given absolute value function in the form :

y - h  =  |x - h|

That is,

y  =  -|x + 1| + 3

Subtract 3 from each side.

y - 3  =  -|x + 1|

To get the vertex, equate (x + 1) and (y - 3) to zero.

x + 1  =  0  and  y - 3  =  0

x  =  -1  and  y  =  3

Therefore, the vertex is

(-1, 3)

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

So, the graph of the given absolute value function is Example 11 :

Graph the absolute value function given below.

y  =  -|x| + 4

Solution :

Write the given absolute value function in the form

y - h  =  |x - h|

That is,

y  =  -|x| + 4

Subtract 4 from each side.

y - 4  =  -|x|

To get the vertex, equate x and (y - 4) to zero.

x  =  0  and  y - 4  =  0

x  =  0  and  y  =  4

Therefore, the vertex is

(0, 4)

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

So, the graph of the given absolute value function is Example 12 :

Graph the absolute value function given below.

y  =  -|x + 1| - 1

Solution :

Write the given absolute value function in the form :

y - h  =  |x - h|

That is,

y  =  -|x + 1| - 1

y + 1  =  -|x + 1|

To get the vertex, equate (x + 1) and (y + 1) to zero.

x + 1  =  0  and  y + 1  =  0

x  =  -1  and  y  =  -1

Therefore, the vertex is

(-1, -1)

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

So, the graph of the given absolute value function is  After having gone through the stuff given above, we hope that the students would have understood, how to graph absolute value function.

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